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For any algebraic expressions S and T and any positive real number b , b , where b â 1 , b â 1 , log b S = log b T log b S = log b T if and only if S = T . S = T . | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
If A = A 0 e k t , A = A 0 e k t , k < 0 , k < 0 , the half-life is t = â ln ( 2 ) k . t = â ln ( 2 ) k . | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
t = ln ( A A 0 ) â 0.000121 . t = ln ( A A 0 ) â 0.000121 . A 0 A 0 is the amount of carbon-14 when the plant or animal died A A is the amount of carbon-14 remaining today t t is the age of the fossil in years | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
If A = A 0 e k t , A = A 0 e k t , k > 0 , k > 0 , the doubling time is t = ln 2 k t = ln 2 k | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
T ( t ) = A e k t + T s , T ( t ) = A e k t + T s , where T s T s is the ambient temperature, A = T ( 0 ) â T s , A = T ( 0 ) â T s , and k k is the continuous rate of cooling. | https://openstax.org/books/college-algebra-2e/pages/6-key-equations |
annual percentage rate (APR) : the yearly interest rate earned by an investment account, also callednominal rate | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
carrying capacity : in a logistic model, the limiting value of the output | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
change-of-base formula : a formula for converting a logarithm with any base to a quotient of logarithms with any other base. | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
common logarithm : the exponent to which 10 must be raised to getx;x;log10(x)log10(x)is written simply aslog(x).log(x). | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
compound interest : interest earned on the total balance, not just the principal | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
doubling time : the time it takes for a quantity to double | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
exponential growth : a model that grows by a rate proportional to the amount present | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
extraneous solution : a solution introduced while solving an equation that does not satisfy the conditions of the original equation | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
half-life : the length of time it takes for a substance to exponentially decay to half of its original quantity | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
logarithm : the exponent to whichbbmust be raised to getx;x;writteny=logb(x)y=logb(x) | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
logistic growth model : a function of the formf(x)=c1+aeâbxf(x)=c1+aeâbxwherec1+ac1+ais the initial value,ccis the carrying capacity, or limiting value, andbbis a constant determined by the rate of growth | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
natural logarithm : the exponent to which the numbereemust be raised to getx;x;loge(x)loge(x)is written asln(x).ln(x). | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
Newtonâs Law of Cooling : the scientific formula for temperature as a function of time as an objectâs temperature is equalized with the ambient temperature | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
nominal rate : the yearly interest rate earned by an investment account, also calledannual percentage rate | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
order of magnitude : the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
power rule for logarithms : a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
product rule for logarithms : a rule of logarithms that states that the log of a product is equal to a sum of logarithms | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
quotient rule for logarithms : a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms | https://openstax.org/books/college-algebra-2e/pages/6-key-terms |
A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the same set of axes. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the second equation. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together. SeeExample 5,Example 6, andExample 7. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect. SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The solution to a system of dependent equations will always be true because both equations describe the same line. SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit. SeeExample 10andExample 11. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
A solution set is an ordered triple{(x,y,z)}{(x,y,z)}that represents the intersection of three planes in space. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. SeeEx... | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Systems of three equations in three variables are useful for solving many different types of real-world problems. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the circle; (3) two solutions, the line intersects the circle in two points. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle:(1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points... | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade the region containing the solution set. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
DecomposeP(x)Q(x)P(x)Q(x)by writing the partial fractions asAa1x+b1+Ba2x+b2.Aa1x+b1+Ba2x+b2.Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The decomposition ofP(x)Q(x)P(x)Q(x)with repeated linear factors must account for the factors of the denominator in increasing powers. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The decomposition ofP(x)Q(x)P(x)Q(x)with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as inAx+Bx+C(ax2+bx+c).Ax+Bx+C(ax2+bx+c).SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
In the decomposition ofP(x)Q(x),P(x)Q(x),whereQ(x)Q(x)has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers asAx+B(ax2+bx+c)+A2x+B2(ax2+bx+c)2+â¯+Anx+Bn(ax2+bx+c)n.Ax+B(ax2+bx+c)+A2x+B2(ax2+bx+c)2+â¯... | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
A matrix is a rectangular array of numbers. Entries are arranged in rows and columns. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The dimensions of a matrix refer to the number of rows and the number of columns. A3Ã23Ã2matrix has three rows and two columns. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. SeeExample 2,Example 3,Example 4, andExample 5. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Scalar multiplication involves multiplying each entry in a matrix by a constant. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Scalar multiplication is often required before addition or subtraction can occur. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Multiplying matrices is possible when inner dimensions are the sameâthe number of columns in the first matrix must match the number of rows in the second. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The product of two matrices,AAandB,B,is obtained by multiplying each entry in row 1 ofAAby each entry in column 1 ofB;B;then multiply each entry of row 1 ofAAby each entry in columns 2 ofB,B,and so on. SeeExample 8andExample 9. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Many real-world problems can often be solved using matrices. SeeExample 10. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. SeeExample 11. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
An augmented matrix is one that contains the coefficients and constants of a system of equations. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
A matrix augmented with the constant column can be represented as the original system of equations. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
We can use Gaussian elimination to solve a system of equations. SeeExample 3,Example 4, andExample 5. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Row operations are performed on matrices to obtain row-echelon form. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. SeeExample 7andExample 8. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
A calculator can be used to solve systems of equations using matrices. SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Many real-world problems can be solved using augmented matrices. SeeExample 10andExample 11. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
An identity matrix has the propertyAI=IA=A.AI=IA=A.SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
An invertible matrix has the propertyAAâ1=Aâ1A=I.AAâ1=Aâ1A=I.SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Use matrix multiplication and the identity to find the inverse of a2Ã22Ã2matrix. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The multiplicative inverse can be found using a formula. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Another method of finding the inverse is by augmenting with the identity. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
We can augment a3Ã33Ã3matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Write the system of equations asAX=B,AX=B,and multiply both sides by the inverse ofA:Aâ1AX=Aâ1B.A:Aâ1AX=Aâ1B.SeeExample 7andExample 8. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
We can also use a calculator to solve a system of equations with matrix inverses. SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The determinant for[abcd][abcd]isadâbc.adâbc.SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Cramerâs Rule replaces a variable column with the constant column. Solutions arex=DxD,y=DyD.x=DxD,y=DyD.SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
To find the determinant of a 3Ã3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
To solve a system of three equations in three variables using Cramerâs Rule, replace a variable column with the constant column for each desired solution:x=DxD,y=DyD,z=DzD.x=DxD,y=DyD,z=DzD.SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Cramerâs Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. SeeExample 5andExample 6. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
Certain properties of determinants are useful for solving problems. For example:If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.When two rows are interchanged, the determinant changes sign.If either two rows or two columns are identical, the determinant eq... | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
When two rows are interchanged, the determinant changes sign. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
If either two rows or two columns are identical, the determinant equals zero. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
The determinant of an inverse matrixAâ1Aâ1is the reciprocal of the determinant of the matrixA.A. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
If any row or column is multiplied by a constant, the determinant is multiplied by the same factor. SeeExample 7andExample 8. | https://openstax.org/books/college-algebra-2e/pages/7-key-concepts |
I 2 = [ 1 0 0 1 ] I 2 = [ 1 0 0 1 ] | https://openstax.org/books/college-algebra-2e/pages/7-key-equations |
I 3 = [ 1 0 0 0 1 0 0 0 1 ] I 3 = [ 1 0 0 0 1 0 0 0 1 ] | https://openstax.org/books/college-algebra-2e/pages/7-key-equations |
A â1 = 1 a d â b c [ d â b â c a ] , where a d â b c â 0 A â1 = 1 a d â b c [ d â b â c a ] , where a d â b c â 0 | https://openstax.org/books/college-algebra-2e/pages/7-key-equations |
addition method : an algebraic technique used to solve systems of linear equations in which the equations are added in a way that eliminates one variable, allowing the resulting equation to be solved for the remaining variable; substitution is then used to solve for the first variable | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
augmented matrix : a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
break-even point : the point at which a cost function intersects a revenue function; where profit is zero | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
coefficient matrix : a matrix that contains only the coefficients from a system of equations | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
column : a set of numbers aligned vertically in a matrix | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
consistent system : a system for which there is a single solution to all equations in the system and it is an independent system, or if there are an infinite number of solutions and it is a dependent system | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
cost function : the function used to calculate the costs of doing business; it usually has two parts, fixed costs and variable costs | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
Cramerâs Rule : a method for solving systems of equations that have the same number of equations as variables using determinants | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
dependent system : a system of linear equations in which the two equations represent the same line; there are an infinite number of solutions to a dependent system | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
determinant : a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
entry : an element, coefficient, or constant in a matrix | https://openstax.org/books/college-algebra-2e/pages/7-key-terms |
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