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Step 4.Graph the solution | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 5.Write the solution using interval notation | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
compound inequality : A compound inequality is made up of two inequalities connected by the word âandâ or the word âor.â | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-terms |
conditional equation : An equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-terms |
contradiction : An equation that is false for all values of the variable is called a contradiction. A contradiction has no solution. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-terms |
identity : An equation that is true for any value of the variable is called an Identity. The solution of an identity is all real numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-terms |
linear equation : A linear equation is an equation in one variable that can be written, whereaandbare real numbers andaâ0,aâ0,asax+b=0.ax+b=0. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-terms |
solution of an equation : A solution of an equation is a value of a variable that makes a true statement when substituted into the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-terms |
y = m x + b y = m x + b | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-concepts |
y â y 1 = m ( x â x 1 ) y â y 1 = m ( x â x 1 ) | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-concepts |
y â y 1 = m ( x â x 1 ) y â y 1 = m ( x â x 1 ) | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-concepts |
boundary line : The line with equationAx+By=CAx+By=Cis the boundary line that separates the region whereAx+By>CAx+By>Cfrom the region whereAx+By<C.Ax+By<C. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
domain of a relation : The domain of a relation is all thex-values in the ordered pairs of the relation. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
function : A function is a relation that assigns to each element in its domain exactly one element in the range. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
horizontal line : A horizontal line is the graph of an equation of the formy=b.y=b.The line passes through they-axis at(0,b).(0,b). | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
intercepts of a line : The points where a line crosses thex-axis and they-axis are called the intercepts of the line. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
linear equation : An equation of the formAx+By=C,Ax+By=C,whereAandBare not both zero, is called a linear equation in two variables. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
linear inequality : A linear inequality is an inequality that can be written in one of the following forms:Ax+By>C,Ax+Byâ¥C,Ax+By<C,Ax+By>C,Ax+Byâ¥C,Ax+By<C,orAx+Byâ¤C,Ax+Byâ¤C,whereAandBare not both zero. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
mapping : A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
ordered pair : An ordered pair,(x,y)(x,y)gives the coordinates of a point in a rectangular coordinate system. The first number is thex-coordinate. The second number is they-coordinate. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
origin : The point(0,0)(0,0)is called the origin. It is the point where thex-axis andy-axis intersect. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
parallel lines : Parallel lines are lines in the same plane that do not intersect. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
perpendicular lines : Perpendicular lines are lines in the same plane that form a right angle. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
point-slope form : The point-slope form of an equation of a line with slopemand containing the point(x1,y1)(x1,y1)isyây1=m(xâx1).yây1=m(xâx1). | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
range of a relation : The range of a relation is all they-values in the ordered pairs of the relation. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
relation : A relation is any set of ordered pairs,(x,y).(x,y).All thex-values in the ordered pairs together make up the domain. All they-values in the ordered pairs together make up the range. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
solution of a linear equation in two variables : An ordered pair(x,y)(x,y)is a solution of the linear equationAx+By=C,Ax+By=C,if the equation is a true statement when thex- andy-values of the ordered pair are substituted into the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
solution to a linear inequality : An ordered pair(x,y)(x,y)is a solution to a linear inequality if the inequality is true when we substitute the values ofxandy. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
standard form of a linear equation : A linear equation is in standard form when it is writtenAx+By=C.Ax+By=C. | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
vertical line : A vertical line is the graph of an equation of the formx=a.x=a.The line passes through thex-axis at(a,0).(a,0). | https://openstax.org/books/intermediate-algebra-2e/pages/3-key-terms |
How to solve a system of linear equations by graphing.Step 1.Graph the first equation.Step 2.Graph the second equation on the same rectangular coordinate system.Step 3.Determine whether the lines intersect, are parallel, or are the same line.Step 4.Identify the solution to the system.If the lines intersect, identify th... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 1.Graph the first equation. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 2.Graph the second equation on the same rectangular coordinate system. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 3.Determine whether the lines intersect, are parallel, or are the same line. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 4.Identify the solution to the system.If the lines intersect, identify the point of intersection. This is the solution to the system.If the lines are parallel, the system has no solution.If the lines are the same, the system has an infinite number of solutions. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 5.Check the solution in both equations. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
How to solve a system of equations by substitution.Step 1.Solve one of the equations for either variable.Step 2.Substitute the expression from Step 1 into the other equation.Step 3.Solve the resulting equation.Step 4.Substitute the solution in Step 3 into either of the original equations to find the other variable.Step... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 1.Solve one of the equations for either variable. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 2.Substitute the expression from Step 1 into the other equation. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 3.Solve the resulting equation. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 4.Substitute the solution in Step 3 into either of the original equations to find the other variable. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 5.Write the solution as an ordered pair. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 6.Check that the ordered pair is a solution tobothoriginal equations. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
How to solve a system of equations by elimination.Step 1.Write both equations in standard form. If any coefficients are fractions, clear them.Step 2.Make the coefficients of one variable opposites.Decide which variable you will eliminate.Multiply one or both equations so that the coefficients of that variable are oppos... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 1.Write both equations in standard form. If any coefficients are fractions, clear them. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 2.Make the coefficients of one variable opposites.Decide which variable you will eliminate.Multiply one or both equations so that the coefficients of that variable are opposites. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 3.Add the equations resulting from Step 2 to eliminate one variable. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 4.Solve for the remaining variable. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 5.Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 6.Write the solution as an ordered pair. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 7.Check that the ordered pair is a solution tobothoriginal equations.Choose the Most Convenient Method to Solve a System of Linear EquationsGraphingâââââSubstitutionâââââââEliminationâââââââUse when you need apicture of the situation.Use when one equation isalready solved or c... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
How To Solve Applications with Systems of EquationsStep 1.Readthe problem. Make sure all the words and ideas are understood.Step 2.Identifywhat we are looking for.Step 3.Namewhat we are looking for. Choose variables to represent those quantities.Step 4.Translateinto a system of equations.Step 5.Solvethe system of equat... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 3.Namewhat we are looking for. Choose variables to represent those quantities. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 4.Translateinto a system of equations. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 5.Solvethe system of equations using good algebra techniques. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Cost function:The cost function is the cost to manufacture each unit timesx, the number of units manufactured, plus the fixed costs.C(x)=(cost per unit)·x+fixed costsC(x)=(cost per unit)·x+fixed costs | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Revenue:The revenue function is the selling price of each unit timesx, the number of units sold.R(x)=(sellingpriceperunit)·xR(x)=(sellingpriceperunit)·x | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Break-even point:The break-even point is when the revenue equals the costs.C(x)=R(x)C(x)=R(x) | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Linear Equation in Three Variables:A linear equation with three variables, wherea, b, c, anddare real numbers anda, b,andcare not all 0, is of the formax+by+cz=dax+by+cz=dEvery solution to the equation is an ordered triple,(x,y,z)(x,y,z)that makes the equation true. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
How to solve a system of linear equations with three variables.Step 1.Write the equations in standard formIf any coefficients are fractions, clear them.Step 2.Eliminate the same variable from two equations.Decide which variable you will eliminate.Work with a pair of equations to eliminate the chosen variable.Multiply o... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 1.Write the equations in standard formIf any coefficients are fractions, clear them. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 2.Eliminate the same variable from two equations.Decide which variable you will eliminate.Work with a pair of equations to eliminate the chosen variable.Multiply one or both equations so that the coefficients of that variable are opposites.Add the equations resulting from Step 2 to eliminate one variable | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 3.Repeat Step 2 using two other equations and eliminate the same variable as in Step 2. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 4.The two new equations form a system of two equations with two variables. Solve this system. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 5.Use the values of the two variables found in Step 4 to find the third variable. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 6.Write the solution as an ordered triple. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 7.Check that the ordered triple is a solution toall threeoriginal equations. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Matrix:A matrix is a rectangular array of numbers arranged in rows and columns. A matrix withmrows andncolumns hasordermÃn.mÃn.The matrix on the left below has 2 rows and 3 columns and so it has order2Ã3.2Ã3.We say it is a 2 by 3 matrix.Each number in the matrix is called anelementorentryin the matrix. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Row Operations:In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.Interchange any two rowsMultiply a row by any real number except 0Add a nonzero multiple of one row to another row | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Interchange any two rows | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Multiply a row by any real number except 0 | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Add a nonzero multiple of one row to another row | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Row-Echelon Form:For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
How to solve a system of equations using matrices.Step 1.Write the augmented matrix for the system of equations.Step 2.Using row operations get the entry in row 1, column 1 to be 1.Step 3.Using row operations, get zeros in column 1 below the 1.Step 4.Using row operations, get the entry in row 2, column 2 to be 1.Step 5... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 1.Write the augmented matrix for the system of equations. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 2.Using row operations get the entry in row 1, column 1 to be 1. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 3.Using row operations, get zeros in column 1 below the 1. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 4.Using row operations, get the entry in row 2, column 2 to be 1. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 5.Continue the process until the matrix is in row-echelon form. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 6.Write the corresponding system of equations. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 7.Use substitution to find the remaining variables. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 8.Write the solution as an ordered pair or triple. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 9.Check that the solution makes the original equations true. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Determinant:The determinant of any square matrix[abcd],[abcd],wherea, b, c,anddare real numbers, is|abcd|=adâbc|abcd|=adâbc | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Expanding by Minors along the First Row to Evaluate a 3 Ã 3 Determinant:To evaluate a3Ã33Ã3determinant by expanding by minors along the first row, the following pattern: | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Sign Pattern:When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.|+â+â+â+â+||+â+â+â+â+| | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Cramerâs Rule:For the system of equations{a1x+b1y=k1a2x+b2y=k2,{a1x+b1y=k1a2x+b2y=k2,the solution(x,y)(x,y)can be determined byNotice that to form the determinantD, we use take the coefficients of the variables. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
How to solve a system of two equations using Cramerâs rule.Step 1.Evaluate the determinantD, using the coefficients of the variables.Step 2.Evaluate the determinantDx.Dx.Use the constants in place of thexcoefficients.Step 3.Evaluate the determinantDy.Dy.Use the constants in place of theycoefficients.Step 4.Findxandy.... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 1.Evaluate the determinantD, using the coefficients of the variables. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 2.Evaluate the determinantDx.Dx.Use the constants in place of thexcoefficients. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 3.Evaluate the determinantDy.Dy.Use the constants in place of theycoefficients. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 4.Findxandy.x=DxD,x=DxD,y=DyD.y=DyD. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 5.Write the solution as an ordered pair. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 6.Check that the ordered pair is a solution tobothoriginal equations. | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 7.Dependent and Inconsistent Systems of Equations:For any system of equations, where thevalue of the determinantD=0,D=0,Value of determinantsType of systemSolutionD=0andDx,DyandDzare all zeroconsistent and dependentinfinitely many solutionsD=0andDx,DyandDzare not all zeroinconsistentno solutionValue of determinant... | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
Step 8.Test for Collinear Points:Three points(x1,y1),(x1,y1),(x2,y2),(x2,y2),and(x3,y3)(x3,y3)are collinear if and only if|x1y11x2y21x3y31|=0|x1y11x2y21x3y31|=0 | https://openstax.org/books/intermediate-algebra-2e/pages/4-key-concepts |
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