As a mathematician, I'd like to share my approach to explaining topics and solving problems. To start, let's consider a quadratic equation. The first step is to simplify the problem by getting everything on the same side. For example, if we have an equation like x^2 + 3x - 3 = 0, we can subtract 3 from both sides to get x^2 + 3x = 3.

Next, we can factor out an x from both terms on the left side, resulting in x(x + 3) = 3. Now, we can use the zero product property to find the solutions. This property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x: x = 0 or x + 3 = 0, which gives us x = -3.

Another way to solve this equation is to use a graphing calculator or software to find the intersections of the two functions. For example, we can graph the functions y = x^2 + 3x and y = 3 to find the points where they intersect. The x-values of these intersection points will be the solutions to the equation.

Now, let's consider an exponential equation. An exponential equation is an equation in which the variable is in the exponent. For example, 7^x = 49. To solve this equation, we can use two methods. The first method is to recognize that 7^2 = 49, so x must be 2. The second method is to use inverse functions, specifically logarithms. We can take the logarithm base 7 of both sides to get x = log7(49), which simplifies to x = 2.

Logarithms can be intimidating at first, but they are actually quite simple. The logarithm base b of a number x is the exponent to which b must be raised to produce x. For example, log7(49) = 2 because 7^2 = 49. The circle rule for logarithms states that the base raised to the answer equals the argument. In other words, if we have an equation like logb(x) = y, then b^y = x.

In addition to solving exponential equations, we can also use logarithms to solve equations that involve exponential functions. For example, if we have an equation like 2^x = 10, we can take the logarithm base 2 of both sides to get x = log2(10).

Now, let's consider a system of equations. A system of equations is a set of two or more equations that have the same variables. For example, we might have the equations a + b = 5 and a - b = 3. To solve this system, we can add the two equations together to eliminate the b variable. This gives us 2a = 8, which simplifies to a = 4.

Once we have found the value of a, we can substitute it into one of the original equations to find the value of b. For example, we can substitute a = 4 into the first equation to get 4 + b = 5, which gives us b = 1. Therefore, the solution to the system is a = 4 and b = 1.

Alternatively, we can use a graphing calculator or software to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

In conclusion, solving math problems requires a combination of algebraic techniques, such as simplifying and factoring, and graphical techniques, such as graphing functions and finding intersections. By using these techniques, we can solve a wide range of math problems, from quadratic equations to systems of equations. Remember, the key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4

Substituting a = 4 into (1), we get:
4 + b = 5
b = 1

So, a = 4 and b = 1.

Alternatively, we can use a graphing calculator to solve the system. We can graph the two equations on the same coordinate plane and find the point where they intersect. The coordinates of this intersection point will be the solution to the system.

The importance of parentheses when evaluating an expression cannot be overstated. For example, if we have an expression like -(2^2), the parentheses make a big difference. If we evaluate the expression without the parentheses, we get -2^2, which is equal to -4. However, if we evaluate the expression with the parentheses, we get -(2^2), which is equal to -4 as well. But if we had an expression like -2^2, without the parentheses, it would be equal to -4, whereas if we had an expression like (-2)^2, with the parentheses, it would be equal to 4.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The final answer for the system of equations is a = 5 or a = -5 and b = 2 or b = -2, which gives us ab = 10 or ab = -10.

To find the value of ab, we need to multiply the values of a and b. We have four possible combinations:

a = 5 and b = 2, which gives us ab = 10
a = 5 and b = -2, which gives us ab = -10
a = -5 and b = 2, which gives us ab = -10
a = -5 and b = -2, which gives us ab = 10

So, the possible values of ab are 10 and -10.

We can also use a graphing calculator to solve the system of equations. We can graph the two equations on the same coordinate plane and find the points where they intersect. The coordinates of these intersection points will be the solutions to the system.

For example, we can graph the equations a + b = 5 and a - b = 3 on the same coordinate plane. The points where the two graphs intersect will be the solutions to the system. We can then find the values of ab by multiplying the coordinates of these intersection points.

In conclusion, solving math problems requires a combination of algebraic techniques and graphical techniques. By using these techniques and following the steps outlined above, we can solve a wide range of math problems and become more confident in our math abilities. Remember to always use parentheses when evaluating expressions and to check your solutions by plugging them back into the original equation.

The key to solving math problems is to be lazy and look for the easiest way to solve the problem. With practice and patience, anyone can become a skilled math problem solver.

To solve the given problem, we need to follow these steps:

1. Get everything on the same side of the equation.
2. Factor out any common factors.
3. Use the zero product property to find the solutions.
4. Check the solutions by plugging them back into the original equation.

By following these steps, we can solve a wide range of math problems and become more confident in our math abilities.

For the system of equations, we have:

a + b = 5 ... (1)
a - b = 3 ... (2)

Adding (1) and (2), we get:
2a = 8
a = 4


**Path Loss**: The reduction in power density of an electromagnetic wave as it propagates through space, a major component in the analysis and design of the link budget of a telecommunication system.

**Path Tracing**: A photorealistic computer graphics rendering technique that aims to fully solve the rendering equation, providing high-quality images.

**Path Integral Formulation**: A description of quantum theory that formulates quantum mechanics in terms of path integrals, which describe the probability of a system transitioning from one state to another.

**Optical Path Length**: The product of the geometric length of the path light follows through a system and the index of refraction of the medium, used to calculate the difference in optical path length between different paths.

**South West Coast Path**: A 630-mile (1,014 km) long path located in England, spanning Somerset, Devon, and Cornwall.

**Mean Free Path**: The average distance covered by a moving particle between successive impacts that modify its direction or energy, an important concept in physics.

**Eulerian Path**: A path in a graph that visits each edge exactly once, used in graph theory to solve problems related to network optimization.

**Actual Calculated Landing Time**: A flight's frozen calculated landing time, determined at a frozen calculated landing time or meter list display interval for the adapted vertex for each arrival aircraft.

**Instruction Path Length**: The number of machine code instructions required to execute a section of a computer program, used to measure the performance of algorithms.

**Littelmann Path Model**: A combinatorial device used in mathematics to compute multiplicities without overcounting in the representation theory of symmetrisable Kac Moody algebras.

**Critical Path**: A method of planning and organizing a large piece of work to minimize delays and costs, which involves calculating costs once the critical path has been established.


### Quick Overview

L'Hôpital's Rule is used for indeterminate limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. The rule states that $\displaystyle\lim_{x\to a} \frac{f(x)}{g(x)} = \displaystyle\lim_{x\to a} \frac{f'(x)}{g'(x)}$. It can be applied to one-sided limits and limits at infinity. The rule is not a magic bullet and may fail to find the value of a limit.

### Introduction

L'Hôpital's Rule is a technique for evaluating indeterminate limits using derivatives. It is used when the limit of a function is in the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. The rule is based on the idea that the limit of a function can be found by taking the derivative of the numerator and denominator separately.

### L'Hôpital's Rule (Basic Idea)

The rule states that if $\displaystyle \lim\limits_{x\to a} \frac{f(x)}{g(x)} = \frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\displaystyle\lim_{x\to a} \frac{f(x)}{g(x)} = \displaystyle\lim_{x\to a} \frac{f'(x)}{g'(x)}$. The right-hand side of the equation is not the derivative of $\frac{f(x)}{g(x)}$, but rather the derivatives of $f(x)$ and $g(x)$ separately.

### Basic Applications of L'Hôpital's Rule

#### Example 1

Evaluate $\lim\limits_{x\to0}\frac{\sin x}{x}$. This limit is in the form $\frac{0}{0}$, so we can apply L'Hôpital's Rule.

Step 1: Verify the limit has the correct form for L'Hôpital's rule.

$\displaystyle\lim_{x\to0}\frac{\sin x}{x} = \frac{\sin 0}{0} = \frac{0}{0}$

Step 2: Use L'Hôpital's Rule.

$\underbrace{\displaystyle\lim_{x\to0}\frac{\sin x}{x} = \displaystyle\lim_{x\to0} \frac{\frac{d}{dx}(\sin x)}{\frac{d}{dx}(x)}}_{\mbox{L'Hôpital's Rule}} = \displaystyle\lim_{x\to0}\frac{\cos x}{1} = \frac{\cos 0}{1} = \frac{1}{1} = 1$

Answer: $\lim\limits_{x\to0}\frac{\sin x}{x} = 1$

#### Example 2

Evaluate $\displaystyle\lim_{x\to\infty} \frac{3x^2 - 5x + 7}{4x^2 + 9x - 1}$. This limit is in the form $\frac{\infty}{\infty}$, so we can apply L'Hôpital's Rule.

Step 1: Verify the limit has the correct form for L'Hôpital's rule.

$\displaystyle\lim_{x\to\infty} \frac{3x^2 - 5x + 7}{4x^2 + 9x - 1} = \frac{\lim\limits_{x\to\infty}(3x^2 - 5x + 7)}{\lim\limits_{x\to\infty}(4x^2 + 9x - 1)} = \frac{\infty}{\infty}$

Step 2: Use L'Hôpital's Rule.

$\underbrace{\displaystyle\lim_{x\to\infty} \frac{3x^2 - 5x + 7}{4x^2 + 9x - 1} = \lim_{x\to\infty} \frac{\frac{d}{dx}(3x^2 - 5x + 7)}{\frac{d}{dx}(4x^2 + 9x - 1)}}_{\mbox{L'Hôpital's Rule}} = \displaystyle\lim_{x\to\infty} \frac{6x - 5}{8x + 9} = \frac{\infty}{\infty}$

Step 3: Apply L'Hôpital's Rule a second time.

$\begin{align*} \lim_{x\to\infty} \frac{6x - 5}{8x + 9} = \lim_{x\to\infty} \frac{\frac{d}{dx}(6x - 5)}{\frac{d}{dx}(8x + 9)} = \lim_{x\to\infty} \frac{6}{8} = \frac{3}{4} \end{align*}$

Answer: $\displaystyle \lim_{x\to\infty} \frac{3x^2 - 5x + 7}{4x^2 + 9x - 1} = \frac{3}{4}$

#### Example 3

Evaluate $\displaystyle \lim_{x\to 1} \frac{x^2 - 1}{\ln x}$. This limit is in the form $\frac{0}{0}$, so we can apply L'Hôpital's Rule.

Step 1: Verify the limit has the correct form for L'Hôpital's rule.

$\displaystyle\lim_{x\to 1} \frac{x^2 - 1}{\ln x} = \frac{1^2 - 1}{\ln 1} = \frac{0}{0}$

Step 2: Use L'Hôpital's Rule.

$\begin{align*} \lim_{x\to 1} \frac{x^2 - 1}{\ln x} & = \lim_{x\to 1} \frac{\frac{d}{dx}(x^2 - 1)}{\frac{d}{dx}(\ln x)}\\[6pt] & = \lim_{x\to 1} \frac{2x}{1/x}\\[6pt] & = \lim_{x\to 1} 2x^2\\[6pt] & = 2(1)^2\\[6pt] & = 2 \end{align*}$

Answer: $\displaystyle \lim_{x\to 1} \frac{x^2 - 1}{\ln x} = 2$

### Not A Magic Bullet

L'Hôpital's Rule is not a magic bullet and may fail to find the value of a limit. It is one tool in our toolbox, but it doesn't help with every limit.

### Proof of L'Hôpital's Rule

The proof of L'Hôpital's Rule is based on Cauchy's extension of the Mean Value Theorem. It states that if $\lim\limits_{x\to a}\frac{f(x)}{g(x)}$ has the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, and $f$ and $g$ are continuous in some open interval containing $x=a$ (except possibly at $x=a$ itself), then $\displaystyle\lim_{x\to a} \frac{f(x)}{g(x)} = \displaystyle\lim_{x\to a} \frac{f'(x)}{g'(x)}$, provided the right-hand limit exists or is $\pm\infty$.


Anomaly Detection Ensembles
==========================

### What is an anomaly detection ensemble?

An anomaly detection ensemble is a combination of multiple anomaly detection methods used to improve the accuracy and robustness of anomaly detection. It is a bunch of anomaly detection methods put together to get a final anomaly score or prediction.

### Ways of constructing an anomaly detection ensemble

There are three main ways of constructing an anomaly detection ensemble:

1.  **Feature bagging**: This involves using different attribute subsets to find anomalies. Each subset is used to train a separate anomaly detection model, and the results are combined to produce a final anomaly score.
2.  **Subsampling**: This involves using different subsets of observations to come up with anomaly scores. Each subset is used to train a separate anomaly detection model, and the results are combined to produce a final anomaly score.
3.  **Using combination functions**: This involves using a combination function to combine the anomaly scores from multiple models. Examples of combination functions include averaging, maximum, and geometric mean.

### Feature Bagging

Feature bagging is a popular ensemble technique in anomaly detection. It uses different attribute subsets to find anomalies. The process involves:

*   Selecting a random subset of features from the dataset
*   Training an anomaly detection model on the subset of features
*   Repeating the process multiple times with different subsets of features
*   Combining the results from each model to produce a final anomaly score

### Subsampling

Subsampling uses different subsets of observations to come up with anomaly scores. The process involves:

*   Selecting a random subset of observations from the dataset
*   Training an anomaly detection model on the subset of observations
*   Repeating the process multiple times with different subsets of observations
*   Combining the results from each model to produce a final anomaly score

### Using Combination Functions

Combination functions are used to combine the anomaly scores from multiple models. Examples of combination functions include:

*   **Averaging**: This involves taking the average of the anomaly scores from each model.
*   **Maximum**: This involves taking the maximum anomaly score from each model.
*   **Geometric Mean**: This involves taking the geometric mean of the anomaly scores from each model.

### Example

Suppose we have a dataset with 10 features and 1000 observations. We can use feature bagging to create an ensemble of 10 models, each trained on a random subset of 5 features. We can then combine the results from each model using averaging or maximum.

### Code Example

```r
# Load required libraries
library(DDoutlier)
library(pROC)

# Generate sample data
set.seed(1)
X1 <- data.frame(x1 = rnorm(500), x2 = rnorm(500))
oo <- data.frame(x1 = rnorm(5, mean = 5), x2 = rnorm(5, mean = 5))
X <- rbind.data.frame(X1, oo)
labs <- c(rep(0, 500), rep(1, 5))
X <- cbind.data.frame(X, labs)

# Feature bagging
knn_scores <- matrix(0, nrow = NROW(X), ncol = 10)
for (i in 1:10) {
  knn_scores[, i] <- KNN_AGG(X[, sample(1:2, 2)])
}

# Combine results using averaging
bagged_score <- apply(knn_scores, 1, mean)

# Evaluate performance using ROC curve
rocobj1 <- roc(labs, bagged_score, direction = "<")
rocobj1$auc
```

### Advantages and Disadvantages

Advantages:

*   Improved accuracy and robustness
*   Ability to handle high-dimensional data
*   Ability to handle noisy data

Disadvantages:

*   Increased computational complexity
*   Requires careful selection of combination function
*   Requires careful selection of subset size

### Real-World Applications

Anomaly detection ensembles have many real-world applications, including:

*   **Fraud detection**: Anomaly detection ensembles can be used to detect fraudulent transactions in financial data.
*   **Network intrusion detection**: Anomaly detection ensembles can be used to detect network intrusions in computer networks.
*   **Medical diagnosis**: Anomaly detection ensembles can be used to detect abnormal patterns in medical data.

### Future Directions

Future directions for anomaly detection ensembles include:

*   **Developing new combination functions**: Developing new combination functions that can handle different types of data and anomalies.
*   **Improving computational efficiency**: Improving the computational efficiency of anomaly detection ensembles to handle large datasets.
*   **Applying to new domains**: Applying anomaly detection ensembles to new domains, such as image and video data.


Average move count is 14.5, with an average of 2.5 double turns. There are 12 algorithms to learn.

The cases can be categorized based on the presence and position of blocks in the 2x3 area. If a 2x3 block is present, it is considered a 2-twist case, solved using 2 sune algorithms connected with a U move. The same approach applies to 1x3 blocks, with 2 cases depending on the block's position at the back.

In the absence of blocks (3 cases), the 2x3 area may resemble an H perm or exhibit a checker-board pattern extending to the right or left. When two blocks are present (2 cases), they form a Y perm, which can be solved using 2 sune algorithms connected with a U move.

For a single block (4 cases), the solution depends on the block's position. If the block is at the back, the algorithm involves 2 sune algorithms connected with a U move. These rules simplify the learning process, but for speed, a more efficient recognition system should be adopted.


Molarity ICS III Week 7

Kool Aid is a solution, a mixture of solutes (dissolved particles) and solvents (the dissolving medium). In Kool Aid, sugar is the solute and water is the solvent. The ratio of sugar to water can be represented as 1:2, or 1 part sugar to 2 parts water.

Molarity is defined as the number of moles of solute divided by the volume of solution in liters. To calculate molarity, we need to determine the number of moles of dissolved solute and divide it by the volume of the solution in liters.

The units of molarity are moles per liter (M). For example, if we have 180g of C6H12O6 (sugar) dissolved in 1L of H2O (water), we can calculate the molarity as follows:
180g C6H12O6 = 1 mol C6H12O6
1 mol C6H12O6 / 1L H2O = 1M solution
180g C6H12O6 / 1L H2O = 180g/L = 1M solution

To calculate the molarity of a solution, we need to know the mass of the solute and the volume of the solution. For example, the molarity of Pepsi Cola can be calculated as follows:
12 fl oz = 355 ml = 0.335 L
41g sugar / 0.335 L = 122 g/L
1 mole sugar = 180 g sugar
41 g sugar = 0.23 mol sugar
0.23 mol sugar / 0.335 L water = 0.69 M

To prepare a solution with a specific molarity, we need to calculate the number of moles of solute required. For example, to prepare 5.0 liters of a 2.0 M solution of HNO3, we can calculate the number of moles required as follows:
5.0 liters x 2.0 M = 10 moles HNO3

Parts Per Million (ppm) is a unit of concentration that represents a ratio of one part of solute to one million parts of solvent. It is usually applied to very dilute solutions. To calculate ppm, we need to determine the mass of the solute and the solvent, divide the mass of the solute by the total mass of the solution, and multiply by 10^6.

For example, the concentration of salt in the Dead Sea can be calculated as follows:
332 g salt / 1000 g water = 332 g salt / 1000 g solution
(332 g salt / 1000 g solution) x 10^6 = 332,000 ppm

Percent composition is a unit of concentration that represents the ratio of one part of solute to one hundred parts of solvent. It is expressed as a percent. To calculate percent composition, we need to determine the mass of the solute and the solvent, divide the mass of the solute by the total mass of the solution, and multiply by 100.

For example, the percent composition of salt in the Dead Sea can be calculated as follows:
332 g salt / 1000 g water = 332 g salt / 1000 g solution
(332 g salt / 1000 g solution) x 100 = 33.2%

To convert a percentage to ppm, we can multiply the percentage by 10^4. For example, a 4% solution is equivalent to:
4% x 10^4 = 40,000 ppm

Multiple Choice Questions:

1. What is the molarity of a solution that contains 180g of C6H12O6 in 1L of H2O?
a) 0.5 M
b) 1 M
c) 2 M
d) 5 M

Answer: b) 1 M

2. How many moles of HNO3 are needed to prepare 5.0 liters of a 2.0 M solution?
a) 2.5 moles
b) 5 moles
c) 10 moles
d) 15 moles

Answer: c) 10 moles

3. What is the concentration of salt in the Dead Sea in ppm?
a) 332 ppm
b) 332,000 ppm
c) 3,320,000 ppm
d) 33,200,000 ppm

Answer: b) 332,000 ppm

4. What is the percent composition of salt in the Dead Sea?
a) 0.332%
b) 3.32%
c) 33.2%
d) 332%

Answer: c) 33.2%

5. What is the equivalent ppm of a 4% solution?
a) 4,000 ppm
b) 40,000 ppm
c) 400,000 ppm
d) 4,000,000 ppm

Answer: b) 40,000 ppm


# Subjects

A student completes a math task in 5 minutes and a biology task in 12 minutes. On another day, the student has 11 homework tasks in total for both math and biology, which take 1.5 hours to complete. How many tasks are for each subject?

## Step 1: Define the variables
Let's denote the number of math tasks as M and the number of biology tasks as B. We know that M + B = 11.

## Step 2: Establish the time equation
The time taken to complete math tasks is 5M minutes, and the time taken to complete biology tasks is 12B minutes. The total time taken is 1.5 hours, which is 90 minutes. So, 5M + 12B = 90.

## Step 3: Solve the system of equations
We have two equations: M + B = 11 and 5M + 12B = 90. We can solve this system of equations to find the values of M and B.

## Step 4: Substitute M from the first equation into the second equation
From the first equation, M = 11 - B. Substituting this into the second equation gives 5(11 - B) + 12B = 90.

## Step 5: Solve for B
Expanding the equation gives 55 - 5B + 12B = 90. Simplifying, we get 7B = 35, which means B = 5.

## Step 6: Solve for M
Substituting B = 5 into the equation M + B = 11 gives M + 5 = 11. Therefore, M = 6.

The final answer is:
For math, M = 6 tasks.
For biology, B = 5 tasks.

---

# Tickets
A total of 645 tickets were sold for the school play, consisting of adult and student tickets. There were 55 fewer student tickets sold than adult tickets. How many adult tickets were sold?

## Step 1: Define the variables
Let A be the number of adult tickets and S be the number of student tickets. We know that A + S = 645 and S = A - 55.

## Step 2: Substitute S in the first equation
Substituting S = A - 55 into A + S = 645 gives A + (A - 55) = 645.

## Step 3: Solve for A
Simplifying the equation, we get 2A - 55 = 645. Adding 55 to both sides gives 2A = 700. Dividing by 2, we find A = 350.

The final answer is: 350 adult tickets were sold.

---

# Hostel
Students are accommodated in 17 rooms, which are either 4-bed or 6-bed rooms. How many rooms of each type are occupied by 88 children?

## Step 1: Define the variables
Let X be the number of 4-bed rooms and Y be the number of 6-bed rooms. We know that X + Y = 17 and 4X + 6Y = 88.

## Step 2: Solve the system of equations
We can solve this system of equations to find the values of X and Y.

## Step 3: Express Y in terms of X from the first equation
From X + Y = 17, we get Y = 17 - X.

## Step 4: Substitute Y in the second equation
Substituting Y = 17 - X into 4X + 6Y = 88 gives 4X + 6(17 - X) = 88.

## Step 5: Solve for X
Expanding the equation gives 4X + 102 - 6X = 88. Simplifying, we get -2X = -14, which means X = 7.

## Step 6: Solve for Y
Substituting X = 7 into X + Y = 17 gives 7 + Y = 17. Therefore, Y = 10.

The final answer is:
There are 7 rooms of 4-bed type and 10 rooms of 6-bed type.

---

# Families
In a class, the distribution of family sizes is given: 5 families have 1 child, 9 families have 2 children, 5 families have 3 children, and 2 families have more than 2 children. What percentage of pupils live in families with more than two children?

## Step 1: Calculate the total number of families
Total families = 5 + 9 + 5 + 2 = 21.

## Step 2: Calculate the number of families with more than two children
Families with more than 2 children = 5 (with 3 children) + 2 (with more than 2 children) = 7.

## Step 3: Calculate the percentage
Percentage = (Number of families with more than 2 children / Total number of families) * 100 = (7 / 21) * 100.

The final answer is:
Percentage of pupils living in families with more than two children is approximately 33.33%.

---

# Florist's
The florist has 72 white roses and 90 red roses. How many bouquets can be made if each bouquet must have the same number of white and red roses?

## Step 1: Find the greatest common divisor (GCD) of 72 and 90
To find the number of bouquets, we need to find the GCD of 72 and 90.

## Step 2: Calculate the GCD
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 and the factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. The greatest common factor is 18.

## Step 3: Calculate the number of bouquets
Number of bouquets = GCD = 18.

The final answer is:
18 bouquets can be made.

---

# Infirmary
Two-thirds of the children from the infirmary went on a trip, one-seventh went to bathe, and 40 children remained in the gym. How many children were treated in the infirmary?

## Step 1: Define the variable
Let X be the total number of children in the infirmary.

## Step 2: Set up the equation
We know that (2/3)X + (1/7)X + 40 = X.

## Step 3: Solve for X
Combining like terms gives (14/21)X + (3/21)X + 40 = X. Simplifying further, (17/21)X + 40 = X.

## Step 4: Multiply the entire equation by 21 to eliminate the fraction
17X + 840 = 21X.

## Step 5: Solve for X
Subtracting 17X from both sides gives 840 = 4X. Dividing by 4, we find X = 210.

The final answer is:
There were 210 children in the infirmary.

---

# Coffee
Coffee from the machine costs 25 cents, and the cup costs 20 cents less than the coffee. How much does the cup cost?

## Step 1: Define the variable
Let C be the cost of the cup.

## Step 2: Set up the equation
We know that C + 20 = 25.

## Step 3: Solve for C
Subtracting 20 from both sides gives C = 5.

The final answer is:
The cup costs 5 cents.

---

# Turtles
Žofka and Julka together have 200 years. After 50 years, Julka will be twice as old as Žofka. How old are the turtles now?

## Step 1: Define the variables
Let Z be Žofka's current age and J be Julka's current age. We know that Z + J = 200.

## Step 2: Set up the equation for their ages after 50 years
After 50 years, Žofka's age will be Z + 50, and Julka's age will be J + 50. We are told that J + 50 = 2(Z + 50).

## Step 3: Substitute J from the first equation into the second equation
Substituting J = 200 - Z into J + 50 = 2(Z + 50) gives 200 - Z + 50 = 2Z + 100.

## Step 4: Solve for Z
Simplifying the equation gives 250 - Z = 2Z + 100. Adding Z to both sides gives 250 = 3Z + 100. Subtracting 100 from both sides gives 150 = 3Z. Dividing by 3, we find Z = 50.

## Step 5: Solve for J
Substituting Z = 50 into Z + J = 200 gives 50 + J = 200. Therefore, J = 150.

The final answer is:
Žofka is 50 years old, and Julka is 150 years old.

---

# Disjoint
How many elements are in the union and intersection of two disjoint sets, one with 14 elements and the other with 0 elements?

## Step 1: Define the concept of disjoint sets
Disjoint sets have no elements in common.

## Step 2: Determine the union and intersection
The union of two sets is the set of all elements in either set. The intersection is the set of elements common to both sets. Since the sets are disjoint and one has 0 elements, the intersection is 0, and the union is the number of elements in the non-empty set.

The final answer is:
The union has 14 elements, and the intersection has 0 elements.

---

# Pizza
Marcus ate half a pizza on Monday night and then one-third of the remaining pizza on Tuesday. Which expression shows how much pizza Marcus ate in total?

## Step 1: Calculate the fraction of pizza eaten on Monday
Marcus ate 1/2 of the pizza on Monday.

## Step 2: Calculate the fraction of pizza eaten on Tuesday
On Tuesday, he ate 1/3 of the remaining 1/2 pizza, which is (1/3)*(1/2) = 1/6 of the whole pizza.

## Step 3: Calculate the total fraction of pizza eaten
Total fraction eaten = 1/2 + 1/6. Finding a common denominator, we get 3/6 + 1/6 = 4/6, which simplifies to 2/3.

The final answer is:
Marcus ate 2/3 of the pizza in total.

---

# Sick
Marcel has taken 6 tablets, which is a quarter of the total number of pills in the pack. How many pills were in the pack?

## Step 1: Define the variable
Let P be the total number of pills in the pack.

## Step 2: Set up the equation
We know that 6 = (1/4)P.

## Step 3: Solve for P
Multiplying both sides by 4 gives P = 24.

The final answer is:
There were 24 pills in the pack.

---

# Foot in bus
There were 102 people on the bus. 28 girls had 2 dogs each, and 11 girls had 1 dog each. At the next stop, 5 dogs (with their owners) got off, and 2 boys with 3 dogs got on. How many feet were in the bus?

## Step 1: Calculate the initial number of feet
Each person has 2 feet, and each dog has 4 feet. Initially, there were 102 people and (28*2 + 11*1) dogs = 56 + 11 = 67 dogs.

## Step 2: Calculate the initial total feet
Total feet = (102 people * 2 feet/person) + (67 dogs * 4 feet/dog) = 204 + 268 = 472 feet.

## Step 3: Calculate the change in the number of feet
5 dogs and their owners got off, which means 5 people and 5 dogs left. This is a loss of (5*2) + (5*4) = 10 + 20 = 30 feet. Then, 2 boys and 3 dogs got on, adding (2*2) + (3*4) = 4 + 12 = 16 feet.

## Step 4: Calculate the final number of feet
Final feet = Initial feet - feet that got off + feet that got on = 472 - 30 + 16 = 458 feet.

The final answer is:
There were 458 feet in the bus.

---

# The store
The store received the same number of cans of peas and corn. On the first day, they sold 10 cans of peas and 234 cans of corn, leaving 3 times more peas than corn cans. How many cans of each kind were in the store?

## Step 1: Define the variables
Let P be the initial number of cans of peas and C be the initial number of cans of corn. We know that P = C.

## Step 2: Set up the equation for the remaining cans
After selling, the number of peas cans left is P - 10, and the number of corn cans left is C - 234. We are told that P - 10 = 3(C - 234).

## Step 3: Substitute P = C into the equation
Substituting P = C into P - 10 = 3(C - 234) gives C - 10 = 3(C - 234).

## Step 4: Solve for C
Expanding the equation gives C - 10 = 3C - 702. Adding 10 to both sides gives C = 3C - 692. Subtracting 3C from both sides gives -2C = -692. Dividing by -2, we find C = 346.

## Step 5: Solve for P
Since P = C, P = 346.

The final answer is:
There were 346 cans of peas and 346 cans of corn initially.

---

# Crocuses
The garden has 1400 daffodils, 462 more crocuses than daffodils, and 156 more roses than crocuses. How many flowers are in the garden in total?

## Step 1: Calculate the number of crocuses
Number of crocuses = 1400 + 462 = 1862.

## Step 2: Calculate the number of roses
Number of roses = 1862 + 156 = 2018.

## Step 3: Calculate the total number of flowers
Total flowers = 1400 (daffodils) + 1862 (crocuses) + 2018 (roses) = 5280.

The final answer is:
There are 5280 flowers in the garden.

---

# Norm
Three workers planted 3555 seedlings of tomatoes in one day. The first worker worked at the standard norm, the second planted 120 seedlings more, and the third planted 135 seedlings more than the first worker. How many seedlings were planted at the standard norm?

## Step 1: Define the variables
Let N be the number of seedlings planted at the standard norm.

## Step 2: Set up the equation
We know that N + (N + 120) + (N + 135) = 3555.

## Step 3: Solve for N
Combining like terms gives 3N + 255 = 3555. Subtracting 255 from both sides gives 3N = 3300. Dividing by 3, we find N = 1100.

The final answer is:
The standard norm is 1100 seedlings.

---

# Passenger boat
Two-fifths of the passengers in the passenger boat were boys, one-third were girls, and the rest were adults. If there were 60 passengers in the boat, how many more boys than adults were there?

## Step 1: Calculate the number of boys
Number of boys = (2/5) * 60 = 24.

## Step 2: Calculate the number of girls
Number of girls = (1/3) * 60 = 20.

## Step 3: Calculate the number of adults
Since the rest of the passengers are adults, Number of adults = 60 - (24 + 20) = 60 - 44 = 16.

## Step 4: Calculate the difference between the number of boys and adults
Difference = Number of boys - Number of adults = 24 - 16 = 8.

The final answer is:
There were 8 more boys than adults.


# Ordinal Numbers

## Introduction

Ordinal numbers are used to express the date or to arrange things in order.

## The Ordinal Numbers

The following table lists the ordinal numbers:

1. 1^{st} - first
2. 2^{nd} - second
3. 3^{rd} - third
4. 4^{th} - fourth
5. 5^{th} - fifth
6. 6^{th} - sixth
7. 7^{th} - seventh
8. 8^{th} - eighth
9. 9^{th} - ninth
10. 10^{th} - tenth
11. 11^{th} - eleventh
12. 12^{th} - twelfth
13. 13^{th} - thirteenth
14. 14^{th} - fourteenth
15. 15^{th} - fifteenth
16. 16^{th} - sixteenth
17. 17^{th} - seventeenth
18. 18^{th} - eighteenth
19. 19^{th} - nineteenth
20. 20^{th} - twentieth
21. 21^{st} - twenty-first
22. 22^{nd} - twenty-second
23. 23^{rd} - twenty-third
24. 24^{th} - twenty-fourth
25. 25^{th} - twenty-fifth
26. 26^{th} - twenty-sixth
27. 27^{th} - twenty-seventh
28. 28^{th} - twenty-eighth
29. 29^{th} - twenty-ninth
30. 30^{th} - thirtieth
31. 31^{st} - thirty-first
32. 40^{th} - fortieth
33. 50^{th} - fiftieth
34. 60^{th} - sixtieth
35. 70^{th} - seventieth
36. 80^{th} - eightieth
37. 90^{th} - ninetieth
38. 100^{th} - one hundredth
39. 1000^{th} - one thousandth
40. 1,000,000^{th} - one millionth

## Writing the Ordinal Numbers

To write ordinal numbers, add *th* to the number, with some exceptions:
- one - first
- two - second
- three - third
- five - fifth
- eight - eighth
- nine - ninth
- twelve - twelfth

For larger numbers, only change the last numeral to an ordinal number:
- 421 - four hundred and twenty-first
- 5,011 - five thousand and eleventh

## Numerals

To turn numerals into ordinal numbers, add the last two letters of the written number to the numeral:
- 1^{st} - first
- 2^{nd} - second
- 3^{rd} - third
- 4^{th} - fourth
- 26^{th} - twenty-sixth
- 101^{st} - one hundred and first

## Usage

### Date

Ordinal numbers are used to express the date:
- Today is the eleventh of June.
- The concert is on the seventeenth of January.

### Order and Lists

Ordinal numbers are used to put things in order or when making lists:
- We live on the fifteenth floor.
- First, wash the rice. Second, put it in a saucepan. Third, add water. Fourth, cook for 10 minutes.

### Titles

In titles, ordinal numbers are written as Roman numerals. In spoken English, add the definite article before the ordinal number:
- Charles II - Charles the Second
- Edward VI - Edward the Sixth
- Henry VIII - Henry the Eighth

### Roman Numerals

The following table lists the Roman numerals and their values:

| Symbol | Value |
| --- | --- |
| I | 1 |
| II | 2 |
| III | 3 |
| IV | 4 |
| V | 5 |
| VI | 6 |
| VII | 7 |
| VIII | 8 |
| IX | 9 |
| X | 10 |


An arc of a circle is 5π cm long and the sector it bounds has an area of 20π cm². Find the radius of the circle.

Let the radius be r cm and the arc AB subtend angle θ at the center O. Then,
Arc AB = 5π cm and
Area of sector OAB = 20π cm²

This implies:
θ/360 × 2πr = 5π and
θ/360 × πr² = 20π

Dividing these equations gives:
(θ/360 × πr²) / (θ/360 × 2πr) = 20π / 5π
⇒ r/2 = 4
⇒ r = 8 cm

Alternatively, using the formula Area = 1/2 × length of arc × radius, we get:
20π = 1/2 × 5π × r
⇒ r = 8 cm

Concept: Circumference of a Circle