Solving Sstems of Equations and Inequalities. Solving Linear Sstems Using Substitution. Solving Linear Sstems Using Elimination.3 Solving Linear Sstems Using Technolog.4 Solving Sstems of Linear Inequalities SEE the Big Idea Tropical Fish (p. 88) Baseball (p. 87) Coins (p. 8) Pizza Shop (p. 7) Sale Prices (p. 65) Mathematical Thinking: Mathematicall proficient students can appl the mathematics the know to solve problems arising in everda life, societ, and the workplace.
Maintaining Mathematical Proficienc Rewriting Literal Equations (A..E) Example Solve the literal equation 3x 9 = 5 for x. 3x 9 = 5 3x 9 + 9 = 5 + 9 3x = 5 + 9 3x 3 = 5 + 9 3 x = 5 + 3 Write the equation. Add 9 to each side. Simplif. Divide each side b 3. Simplif. The rewritten literal equation is x = 5 + 3. Solve the literal equation for.. 4 4x = 6. 3 + x = 8 3. x = 4 + 6 4. x = 7 5. x = 4 + z + 6 6. + 6x = z Graphing Linear Inequalities in Two Variables (A.3.D) Example Graph x + > 8 in a coordinate plane. Step Graph x + = 8, or = x + 4. Use a dashed line because the inequalit smbol is >. Step Test (, ). 5 3 x + > 8 Write the inequalit. + () >? 8 Substitute. (, ) 4 x 8 Simplif. Step 3 Because (, ) is not a solution, shade the half-plane that does not contain (, ). Graph the inequalit in a coordinate plane. 7. x < 8. x + > 3 9. 3x + 5 8. x 6. x > 4. 7 3. ABSTRACT REASONING Can ou alwas use (, ) as a test point when graphing a linear inequalit in two variables? Explain our reasoning. 57
Mathematical Thinking Mathematicall profi cient students select tools, including real objects, manipulatives, paper and pencil, and technolog as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. (A..C) Using a Graphing Calculator Core Concept Graphing a Sstem of Linear Inequalities You can use a graphing calculator to find all solutions, if the exist, of a sstem of linear inequalities.. Enter the inequalities into a graphing calculator.. Graph the inequalities in an appropriate viewing window, so that the intersection of the half-planes is visible. 3. Find the intersection of the half-planes, which is the graph of all the solutions of the sstem. Using a Graphing Calculator Use a graphing calculator to find the solution, if it exists, of the sstem of linear inequalities. x Inequalit < x + 5 Inequalit SOLUTION The slopes of the boundar lines are not the same, so ou know that the lines intersect. Enter the inequalities into a graphing calculator. Then graph the inequalities in an appropriate viewing window. greater than less than Plot Plot Plot3 Y=X- Y=(/)X+5 Y3= Y4= Y5= Y6= Y7= Find the intersection of the half-planes. Note that an point on the boundar line = x is a solution, and an point on the boundar line = x + 5 is not a solution. One solution is (, ). 8 The solution is the double-shaded region. 8 Monitoring Progress Use a graphing calculator to graph the sstem of linear inequalities. Name one solution, if an, of the sstem.. 3x. x + > 3 3. x 4 > x + 4 6x + < 4x 5 58 Chapter Solving Sstems of Equations and Inequalities
. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.3.A A.3.B Solving Linear Sstems Using Substitution Essential Question How can ou determine the number of solutions of a linear sstem? A linear sstem is consistent when it has at least one solution. A linear sstem is inconsistent when it has no solution. Recognizing Graphs of Linear Sstems Work with a partner. Match each linear sstem with its corresponding graph. Explain our reasoning. Then classif the sstem as consistent or inconsistent. a. x 3 = 3 b. x 3 = 3 c. x 3 = 3 4x + 6 = 6 x + = 5 4x + 6 = 6 A. B. C. 4 x 4 x 4 x Solving Sstems of Linear Equations Work with a partner. Solve each linear sstem b substitution. Then use the graph of the sstem below to check our solution. a. x + = 5 b. x + 3 = c. x + = x = x + = 4 3x + = 4 FORMULATING A PLAN To be proficient in math, ou need to formulate a plan to solve a problem. 4 x 4 Communicate Your Answer 3. How can ou determine the number of solutions of a linear sstem? 4. Suppose ou were given a sstem of three linear equations in three variables. Explain how ou would solve such a sstem b substitution. x x 5. Appl our strateg in Question 4 to solve the linear sstem. x + + z = Equation x z = 3 Equation x + z = Equation 3 Section. Solving Linear Sstems Using Substitution 59
. Lesson What You Will Learn Core Vocabular linear equation in three variables, p. 6 sstem of three linear equations, p. 6 solution of a sstem of three linear equations, p. 6 ordered triple, p. 6 Previous sstem of two linear equations Visualize solutions of sstems of linear equations in three variables. Solve sstems of linear equations in three variables b substitution. Solve real-life problems. Visualizing Solutions of Sstems A linear equation in three variables x,, and z is an equation of the form ax + b + cz = d, where a, b, and c are not all zero. The following is an example of a sstem of three linear equations in three variables. 3x + 4 8z = 3 Equation x + + 5z = Equation 4x + z = Equation 3 A solution of such a sstem is an ordered triple (x,, z) whose coordinates make each equation true. The graph of a linear equation in three variables is a plane in three-dimensional space. The graphs of three such equations that form a sstem are three planes whose intersection determines the number of solutions of the sstem, as shown in the diagrams below. Exactl One Solution The planes intersect in a single point, which is the solution of the sstem. Infinitel Man Solutions The planes intersect in a line. Ever point on the line is a solution of the sstem. The planes could also be the same plane. Ever point in the plane is a solution of the sstem. No Solution There are no points in common with all three planes. 6 Chapter Solving Sstems of Equations and Inequalities
Solving Sstems of Equations b Substitution The substitution method for solving sstems of linear equations in two variables can be extended to solve a sstem of linear equations in three variables. Core Concept Solving a Three-Variable Sstem b Substitution Step Solve one equation for one of its variables. Step Substitute the expression from Step in the other two equations to obtain a linear sstem in two variables. Step 3 Solve the new linear sstem for both of its variables. Step 4 Substitute the values found in Step 3 into one of the original equations and solve for the remaining variable. ANALYZING MATHEMATICAL RELATIONSHIPS The missing x-term in Equation makes it convenient to solve for or z. ANOTHER WAY In Step, ou could also solve Equation for z. When ou obtain a false equation, such as =, in an of the steps, the sstem has no solution. When ou do not obtain a false equation, but obtain an identit such as =, the sstem has infinitel man solutions. Solving a Three-Variable Sstem (One Solution) Solve the sstem b substitution. 3 6z = 6 Equation SOLUTION x + 4z = Equation x + z = Equation 3 Step Solve Equation for. = z New Equation Step Substitute z for in Equations and 3 to obtain a sstem in two variables. x (z ) + 4z = Substitute z for in Equation. x + z = 8 New Equation x + (z ) z = Substitute z for in Equation 3. x + 3z = 6 New Equation 3 Step 3 Solve the new linear sstem for both of its variables. x = 8 z Solve new Equation for x. (8 z) + 3z = 6 Substitute 8 z for x in new Equation 3. z = Solve for z. x = 8 Substitute into new Equation 3 to find x. Step 4 Substitute x = 8 and z = into an original equation and solve for. 3 6z = 6 Write original Equation. 3 6() = 6 Substitute for z. = Solve for. The solution is x = 8, =, and z =, or the ordered triple (8,, ). Check this solution in each of the original equations. Section. Solving Linear Sstems Using Substitution 6
Solving a Three-Variable Sstem (No Solution) Solve the sstem b substitution. 4x + z = 5 Equation SOLUTION 8x + z = Equation x + + 7z = 3 Equation 3 Step Solve Equation for z. z = 4x + + 5 New Equation Step Substitute 4x + + 5 for z in Equations and 3 to obtain a sstem in two variables. 8x + ( 4x + + 5) = Substitute 4x + + 5 for z in Equation. = New Equation Because ou obtain a false equation, ou can conclude that the original sstem has no solution. Solving a Three-Variable Sstem (Man Solutions) Solve the sstem b substitution. 4x + z = Equation SOLUTION 4x + + z = Equation x + 3 3z = 6 Equation 3 Step Solve Equation for. = 4x + z + New Equation Step Substitute 4x + z + for in Equations and 3 to obtain a sstem in two variables. 4x + ( 4x + z + ) + z = Substitute 4x + z + for in Equation. z = New Equation x + 3( 4x + z + ) 3z = 6 Substitute 4x + z + for in Equation 3. 6 = 6 New Equation 3 Because ou obtain the identit 6 = 6, the sstem has infinitel man solutions. Step 3 Describe the solutions of the sstem using an ordered triple. One wa to do this is to substitute for z in Equation to obtain = 4x +. So, an ordered triple of the form (x, 4x +, ) is a solution of the sstem. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem b substitution. Check our solution, if possible.. x + + z = 7. x + z = 4 3. x + 6z = x + 3 z = 5 x z = 4 x + z = 8 x 5 + z = 3 3x + 3 + z = 5x + + 7z = 4. In Example 3, describe the solutions of the sstem using an ordered triple in terms of. 6 Chapter Solving Sstems of Equations and Inequalities
Solving Real-Life Problems Appling Mathematics B LAWN B B B A A A B An amphitheater charges $75 for each seat in Section A, $55 for each seat in Section B, and $3 for each lawn seat. There are three times as man seats in Section B as in Section A. The revenue from selling all 3, seats is $87,. How man seats are in each section of the amphitheater? STAGE SOLUTION Step Write a verbal model for the situation. Number of seats in B, = 3 Number of seats in A, x Number of seats in A, x Number of + seats in B, Number of + lawn seats, z = Total number of seats 75 Number of seats in A, x + 55 Number of seats in B, + 3 Number of lawn seats, z = Total revenue Step Write a sstem of equations. = 3x Equation x + + z = 3, Equation 75x + 55 + 3z = 87, Equation 3 Step 3 Substitute 3x for in Equations and 3 to obtain a sstem in two variables. x + 3x + z = 3, Substitute 3x for in Equation. 4x + z = 3, New Equation 75x + 55(3x) + 3z = 87, Substitute 3x for in Equation 3. 4x + 3z = 87, New Equation 3 Step 4 Solve the new linear sstem for both of its variables. STUDY TIP When substituting to find values of other variables, choose original or new equations that are easiest to use. 4x + 3( 4x + 3,) = 87, z = 4x + 3, Solve new Equation for z. x = 5 Solve for x. Substitute 4x + 3, for z in new Equation 3. = 45 Substitute into Equation to find. z = 7, Substitute into Equation to find z. The solution is x = 5, = 45, and z = 7,, or (5, 45, 7,). So, there are 5 seats in Section A, 45 seats in Section B, and 7, lawn seats. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. WHAT IF? On the first da,, tickets sold, generating $356, in revenue. The number of seats sold in Sections A and B are the same. How man lawn seats are still available? Section. Solving Linear Sstems Using Substitution 63
. Exercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY The solution of a sstem of three linear equations is expressed as a(n).. DIFFERENT WORDS, SAME QUESTION Consider the sstem of linear equations shown. Which is different? Find both answers. Solve the sstem of linear equations. x + 3 = x + + z = 3 x + 3 z = 7 Solve each equation in the sstem for. Find the ordered triple whose coordinates make each equation true. Find the point of intersection of the planes modeled b the linear sstem. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, determine whether the ordered triple is a solution of the sstem. Justif our answer. 3. (4, 5, ) 4. (, 3, 6) x + + 5z = 8 x + + z = 4 x + 3 + z = 9 4x + 3z = 3 x + z = 3 x 5 + z = 3 In Exercises 5 4, solve the sstem b substitution. (See Example.) 5. x = 4 6. x 3 + z = x + = 6 + z = 3 4x 3 + z = 6 z = 5 7. x + = 8. x + z = 3 x + 3 + z = 4 5 z = 3 x + 4z = x + 3 + z = 9. x + 6 + 7z = 35. x + + z = 7x 5 6z = 5x + 5 + 5z = x + = x 4 + z =. x + + z = 4. 3x + + z = 3 5x + 3 + z = 56 7x + 6z = 37 x + z = x + 3z = 4 3. 3x 4 + z = 6 4. x 3 + 6z = x + z = 3 3x + 5z = 3 9x 4 z = 4 x 5 + z = 6 ERROR ANALYSIS In Exercises 5 and 6, describe and correct the error in the first steps of solving the sstem of linear equations. x + z = 3 3x + + z = x z = 5. 6. z = 3x x 3x = x 3 = 9 = x + z x + ( x + z) z = 3 x z = 5 In Exercises 7, solve the sstem b substitution. (See Examples and 3.) 7. + 3z = 3 8. x = z x + + z = 8 x + + z = x + 3 z = 3x + 3 + 6z = 4 9. x + 3z =. x + z = 44 7x + 3 z = x 3 + 5z = 8 4x + 6z = 4 x + z = 4. x + 3 z = 6. x 3 + z = 3x + 6z = 9 x + + z = 6 x + 4 z = 3 3x 9 + 3z = 64 Chapter Solving Sstems of Equations and Inequalities
3. MODELING WITH MATHEMATICS A wholesale store advertises that for $ ou can bu one pound each of peanuts, cashews, and almonds. Cashews cost as much as peanuts and almonds combined. You purchase pounds of peanuts, pound of cashews, and 3 pounds of almonds for $36. What is the price per pound of each tpe of nut? (See Example 4.) 8. MODELING WITH MATHEMATICS Use a sstem of linear equations to model the data in the following newspaper article. Solve the sstem to find how man athletes finished in each place. Lawrence High prevailed in Saturda s track meet with the help of individual-event placers earning a combined 68 points. A first-place finish earns 5 points, a secondplace finish earns 3 points, and a third-place finish earns point. Lawrence had a strong second-place showing, with as man second place finishers as first- and third-place finishers combined. 4. MODELING WITH MATHEMATICS Each ear, votes are cast for the rookie of the ear in a softball league. The voting results for the top three finishers are shown in the table below. How man points is each vote worth? Plaer st place nd place 3rd place Points Plaer 3 5 3 Plaer 5 7 4 8 Plaer 3 5 5 35 5. WRITING Write a linear sstem in three variables for which it is easier to solve for one variable than to solve for either of the other two variables. Explain our reasoning. 6. REPEATED REASONING Using what ou know about solving linear sstems in two and three variables b substitution, plan a strateg for how ou would solve a sstem that has four linear equations in four variables. 7. PROBLEM SOLVING The number of left-handed people in the world is one-tenth the number of right-handed people. The percent of right-handed people is nine times the percent of left-handed people and ambidextrous people combined. What percent of people are ambidextrous? MATHEMATICAL CONNECTIONS In Exercises 9 and 3, write and use a linear sstem to answer the question. 9. The triangle has a perimeter of 65 feet. What are the lengths of sides, m, and n? = m 3 m n = + m 5 3. What are the measures of angles A, B, and C? B A (5A C) A (A + B) C 3. OPEN-ENDED Write a sstem of three linear equations in three variables that has the ordered triple ( 4,, ) as its onl solution. Justif our answer using the substitution method. 3. MAKING AN ARGUMENT A linear sstem in three variables has no solution. Your friend concludes that it is not possible for two of the three equations to have an points in common. Is our friend correct? Explain our reasoning. Section. Solving Linear Sstems Using Substitution 65
33. PROBLEM SOLVING A contractor is hired to build an 36. HOW DO YOU SEE IT? Determine whether the apartment complex. Each 84-square-foot unit has a bedroom, kitchen, and bathroom. The bedroom will be the same size as the kitchen. The owner orders 98 square feet of tile to completel cover the floors of two kitchens and two bathrooms. Determine how man square feet of carpet is needed for each bedroom. BATHROOM sstem of equations that represents the circles has no solution, one solution, or infinitel man solutions. Explain our reasoning. a. b. x x KITCHEN 37. REASONING Consider a sstem of three linear equations in three variables. Describe the possible number of solutions in each situation. BEDROOM Total Area: 84 ft a. The graphs of two of the equations in the sstem are parallel planes. b. The graphs of two of the equations in the sstem intersect in a line. 34. THOUGHT PROVOKING Consider the sstem shown. x 3 + z = 6 x + 4 z = 9 c. The graphs of two of the equations in the sstem are the same plane. a. How man solutions does the sstem have? 38. ANALYZING RELATIONSHIPS Use the integers 3,, and to write a linear sstem that has a solution of (3,, 7). b. Make a conjecture about the minimum number of equations that a linear sstem in n variables can have when there is exactl one solution. x 3 + 3z = x + + z = 3 x 5 + z = 6 35. PROBLEM SOLVING A florist must make 5 identical bridesmaid bouquets for a wedding. The budget is $6, and each bouquet must have flowers. Roses cost $.5 each, lilies cost $4 each, and irises cost $ each. The florist wants twice as man roses as the other two tpes of flowers combined. 39. ABSTRACT REASONING Write a linear sstem to represent the first three pictures below. Use the sstem to determine how man tangerines are required to balance the apple in the fourth picture. Note: The first picture shows that one tangerine and one apple balance one grapefruit. a. Write a sstem of equations to represent this situation, assuming the florist plans to use the maximum budget. b. Solve the sstem to find how man of each tpe of flower should be in each bouquet. 3 4 5 6 7 8 9 3 4 5 6 7 8 9 c. Suppose there is no limitation on the total cost of the bouquets. Does the problem still have exactl one solution? If so, find the solution. If not, give three possible solutions. Reviewing what ou learned in previous grades and lessons Solve the sstem of linear equations b elimination. (Skills Review Handbook) 4. x + 3 = 6 4. x = 3 4. 4x + = 4 43. 4x 3 = 9 x + 6 = 44 66 Chapter 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Maintaining Mathematical Proficienc x = 5 3 4 5 6 7 8 9 3 4 5 6 7 8 9 5x + = 3 5x = 6 Solving Sstems of Equations and Inequalities
. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.3.B Solving Linear Sstems Using Elimination Essential Question How can ou rewrite a linear sstem so that it can be solved using mental math? A linear sstem in row-echelon form has a stair-step pattern with leading coefficients of. A sstem can be written in row-echelon form b producing a series of equivalent sstems. Equivalent sstems have the same solution. Recognizing Graphs of Linear Sstems Work with a partner. Match each linear sstem in row-echelon form with its corresponding graph. Explain our reasoning. a. x + = 4 b. x = c. x + = = = = A. 3 B. 3 C. 3 x 3 x 3 x Writing Linear Sstems in Row-Echelon Form Work with a partner. Match each linear sstem with its equivalent sstem in rowechelon form. Justif our answers. a. = b. x + = 5 c. x + 4 = 4 x 3 = x = 3 3x 5 = 8 REASONING To be proficient in math, ou need to analze mathematical relationships to connect mathematical ideas. A. x + = 5 B. x + = 7 C. x 3 = = = 3 = Communicate Your Answer 3. How can ou rewrite a linear sstem so that it can be solved using mental math? 4. Equivalent sstems are produced using row operations. Describe the row operations ou used in Exploration to produce equivalent sstems. 5. Use row operations to write the linear sstem in three variables in row-echelon form. x + 4z = 6 Equation x + + z = Equation z = Equation 3 Section. Solving Linear Sstems Using Elimination 67
. Lesson What You Will Learn Core Vocabular Gaussian elimination, p. 7 Previous linear equation in three variables sstem of three linear equations solution of a sstem of three linear equations ordered triple sstem of two linear equations substitution ANALYZING MATHEMATICAL RELATIONSHIPS The missing -term in Equation makes a convenient variable to eliminate. Solve sstems of linear equations in three variables b elimination. Solve sstems of linear equations b Gaussian elimination. Solving Sstems of Equations b Elimination The elimination method for solving sstems of linear equations in two variables can also be extended to solve a sstem of linear equations in three variables. Core Concept Solving a Three-Variable Sstem b Elimination Step Eliminate one variable to obtain a linear sstem in two variables. Step Solve the new linear sstem for both of its variables. Step 3 Substitute the values found in Step into one of the original equations and solve for the remaining variable. When ou obtain a false equation, such as =, in an of the steps, the sstem has no solution. When ou do not obtain a false equation, but obtain an identit such as =, the sstem has infinitel man solutions. Solving a Three-Variable Sstem (One Solution) Solve the sstem b elimination. 4x + + 3z = Equation SOLUTION 6x + 5z = 6 Equation 6x + 4z = 3 Equation 3 Step Rewrite the sstem as a linear sstem in two variables. 4x + + 3z = Add times Equation 3 to x + 8z = 6 Equation (to eliminate ). 6x + z = 6 New Equation Step Solve the new linear sstem for both of its variables. 6x + z = 6 Add times Equation 6x 5z = 6 to new Equation (to eliminate x). 6z = z = Solve for z. x = Substitute into new Equation to find x. Step 3 Substitute x = and z = into an original equation and solve for. 6x + 4z = 3 Write original Equation 3. 6( ) + 4() = 3 Substitute for x and for z. = 5 Solve for. The solution is x =, = 5, and z =, or the ordered triple (, 5, ). Check this solution in each of the original equations. 68 Chapter Solving Sstems of Equations and Inequalities
Solving a Three-Variable Sstem (No Solution) Solve the sstem b elimination. x + + z = Equation SOLUTION 5x + 5 + 5z = 3 Equation 4x + 3z = 6 Equation 3 Step Rewrite the sstem as a linear sstem in two variables. 5x 5 5z = Add 5 times Equation 5x + 5 + 5z = 3 to Equation. = 7 Because ou obtain a false equation, the original sstem has no solution. ANOTHER WAY Subtracting Equation from Equation gives z =. After substituting for z in each equation, ou can see that each is equivalent to = x + 3. Solving a Three-Variable Sstem (Man Solutions) Solve the sstem b elimination. x + z = 3 Equation x z = 3 Equation 5x 5 + z = 5 Equation 3 SOLUTION Step Rewrite the sstem as a linear sstem in two variables. x + z = 3 Add Equation to x z = 3 Equation (to eliminate z). x = 6 New Equation x z = 3 Add Equation to 5x 5 + z = 5 Equation 3 (to eliminate z). 6x 6 = 8 New Equation 3 Step Solve the new linear sstem for both of its variables. 6x + 6 = 8 Add 3 times new Equation 6x 6 = 8 to new Equation 3. = Because ou obtain the identit =, the sstem has infinitel man solutions. Step 3 Describe the solutions of the sstem using an ordered triple. One wa to do this is to solve new Equation for to obtain = x + 3. Then substitute x + 3 for in original Equation to obtain z =. So, an ordered triple of the form (x, x + 3, ) is a solution of the sstem. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem b elimination. Check our solution, if possible.. x + z =. x + z = 3. x + + z = 8 3x + z = 7 4x + 4 4z = x + z = 8 x + + 4z = 9 3x + + z = x + + z = 6 4. In Example 3, describe the solutions of the sstem using an ordered triple in terms of. Section. Solving Linear Sstems Using Elimination 69
STUDY TIP The sstem of equations x + + z = 5 z = z = is in row-echelon form. Notice its solution of (,, ) can be easil found. Solving Sstems b Gaussian Elimination Sstems written in row-echelon form can be easil solved using substitution. A sstem in row-echelon form has a stair-step pattern with leading coefficients of. To solve a sstem that is not in row-echelon form, use the operations shown below to rewrite the sstem in its equivalent row-echelon form. This process is called Gaussian elimination, after German mathematician Carl Friedrich Gauss (777 855). Core Concept Operations that Produce Equivalent Sstems Each of the following row operations on a sstem of linear equations produces an equivalent sstem of linear equations.. Interchange two equations.. Multipl one of the equations b a nonzero constant. 3. Add a multiple of one of the equations to another equation to replace the latter equation. Using Gaussian Elimination to Solve a Sstem Solve the sstem b Gaussian elimination. x + 3z = 9 Equation SOLUTION x + 3 = 4 Equation 5 4z = 3 Equation 3 Because the leading coefficient of the first equation is, begin b keeping the x at the upper left position and eliminating the other x-term from the first column. x + 3z = 9 Write Equation. x + 3 = 4 Write Equation. + 3z = 5 Add Equation to Equation. x + 3z = 9 + 3z = 5 5 4z = 3 Adding the first equation to the second equation produces a new second equation. 5 + 5z = 5 Multipl new Equation b 5. 5 4z = 3 Write Equation 3. z = Add new Equation to Equation 3. x + 3z = 9 + 3z = 5 z = Adding 5 times the second equation to the third equation produces a new third equation. Using substitution, ou can conclude that the solution is x =, =, and z =, or the ordered triple (,, ). Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Use Gaussian elimination to solve the sstem of linear equations in Monitoring Progress Question. 7 Chapter Solving Sstems of Equations and Inequalities
. Exercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. WRITING How is solving a linear sstem b elimination similar to solving a linear sstem b substitution?. WRITING Explain how ou know when a linear sstem in three variables has infinitel man solutions. Monitoring Progress and Modeling with Mathematics In Exercises 3 8, solve the sstem b elimination. (See Examples,, and 3.) 3. x + z = 5 4. x + 4 6z = x + + z = x + z = 7 x + 3 z = 9 x + 4z = 5 5. 3x + z = 4 6. 5x + z = 6 6x + 4z = 8 x + + z = x + 3z = x + 4 = 7. x + 3 z = 8. x 3 + z = 6 x + z = x + z = 5 3x + 3z = 7x + 8 6z = 3 3. x + z = 3 4. x + + 3z = 4 x + 4 z = 6 3x + z = x + z = 6 x 4z = 4 5. x + z = 3 6. 4x + + 5z = 5 x + z = 8x + + z = 6x 3 z = 7 x z = 7. MODELING WITH MATHEMATICS Three orders are placed at a pizza shop. Two small pizzas, a liter of soda, and a salad cost $4; one small pizza, a liter of soda, and three salads cost $5; and three small pizzas, a liter of soda, and two salads cost $. How much does each item cost? ERROR ANALYSIS In Exercises 9 and, describe and correct the error in the first step of solving the sstem of linear equations b elimination. 4x + z = 8 x + + z = 3x + 3 4z = 44 9.. 4x + z = 8 4x + + z = + 3z = 7 x 3 + 6z = 8 3x + 3 4z = 44 5x + z= 6 8. MODELING WITH MATHEMATICS Sam s Furniture Store places the following advertisement in the local newspaper. Write a sstem of equations for the three combinations of furniture. What is the price of each piece of furniture? Explain. SAM S Furniture Store In Exercises 6, solve the sstem b Gaussian elimination. (See Example 4.). x + z = 4. x z = 5 3x + + 4z = 7 4x + 5 + z = x + 5 + z = 8 x 4 + 3z = Sofa and love seat Sofa and two chairs Sofa, love seat, and one chair Section. Solving Linear Sstems Using Elimination 7
9. MODELING WITH MATHEMATICS A pla is performed for a crowd of 4 people. Adult tickets cost $ each, student tickets cost $5 each, and tickets for children cost $3.5 each. The revenue for the concert is $784. There are 4 more children at the concert than students. How man of each tpe of ticket are sold? PLAYHOUSE TICKETS ADULT: $ STUDENT: $5 CHILD: $3.5. MODELING WITH MATHEMATICS A stadium has, seats, divided into box seats, lower-deck seats, and upper-deck seats. Box seats sell for $, lowerdeck seats sell for $8, and upper-deck seats sell for $5. When all the seats for a game are sold, the total revenue is $7,. The stadium has four times as man upper-deck seats as box seats. Find the number of lower-deck seats in the stadium.. COMPARING METHODS Determine whether ou would use elimination or Gaussian elimination to solve each sstem. Explain our reasoning. a. x + + 5z = b. 3x + 3z = x + z = 7x + 5z = 4 x + 4 3z = 4 x + 4 4z = 6. HOW DO YOU SEE IT? Consider the diagram below. Write a sstem of linear equations in three variables that models the situation, where the variables represent the numbers of each tpe of coin. 3. CRITICAL THINKING Find the values of a, b, and c so that the linear sstem shown has (,, 3) as its onl solution. Explain our reasoning. x + 3z = a x + z = b x + 3 z = c 4. THOUGHT PROVOKING Does the sstem of linear equations have more than one solution? Justif our answer. 4x + + z = x + 3z = x 4 z = 5. OPEN-ENDED Consider the sstem of linear equations below. Choose nonzero values for a, b, and c so the sstem satisfies the given condition. Explain our reasoning. x + + z = ax + b + cz = x + z = 4 a. The sstem has no solution. b. The sstem has exactl one solution. c. The sstem has infinitel man solutions. 6. PROBLEM SOLVING You spend $4 on pounds of apples. You purchase pounds of golden delicious apples for $.5 per pound. Red delicious apples cost $. per pound and empire apples cost $.5 per pound. Write a sstem of equations in row-echelon form that represents this situation. How man pounds of each tpe of apple did ou purchase? Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Use a graphing calculator to solve the sstem of linear equations. (Skills Review Handbook) 7. 3x + = 7 8. 4x 3 = 7 9. x + 8 = 37 x + = x + 7 = 47 x + 4 = 39 7 Chapter Solving Sstems of Equations and Inequalities
.. What Did You Learn? Core Vocabular linear equation in three variables, p. 6 sstem of three linear equations, p. 6 solution of a sstem of three linear equations, p. 6 ordered triple, p. 6 Gaussian elimination, p. 7 Core Concepts Section. Visualizing Solutions of Sstems, p. 6 Solving a Three-Variable Sstem b Substitution, p. 6 Section. Solving a Three-Variable Sstem b Elimination, p. 69 Operations that Produce Equivalent Sstems, p. 7 Mathematical Thinking. How did ou use the information in the newspaper article in Exercise 8 on page 65 to write a sstem of three linear equations?. Explain the strateg ou used to choose the values for a, b, and c in Exercise 5 on page 7. Stud Skills Ten Steps for Test Taking. As soon as ou get the test, turn it over and write down an formulas, calculations, and rules that ou still have trouble remembering.. Preview the test and mark the questions ou know how to do easil. These are the problems ou should do first. 3. As ou preview the test, ou ma have remembered other information. Write this information on the back of the test. 4. Based on how man points each question is worth, decide on a progress schedule. You should alwas have more than half of the test done before half the time has elapsed. 5. Solve the problems ou marked while previewing the test. 6. Skip the problems that ou suspect will give ou trouble. 7. After solving all the problems that ou know how to do easil, go back and reread the problems ou skipped. 8. Tr our best at the remaining problems. Even if ou cannot solve a problem, ou ma be able to get partial credit for a few correct steps. 9. Review the test, looking for an careless errors ou ma have made.. The test is not a race against the other students. Use all the allowed test time. 73
.. Quiz. Determine whether each ordered triple is a solution of the sstem of equations shown. Explain our reasoning. (Section.) a. (,, ) b. ( 3,, 5) c. (,, 4) d. (4,, 6) x + + z = 4 Equation 3x + + 3z = Equation x z = 4 Equation 3 Solve the sstem b substitution. (Section.). x + 3z = 3. x + 4z = 4. x 3 + z = = z x + + z = x 3z = x + = x + z = 5 x + z = Solve the sstem b elimination. (Section.) 5. x + 4 z = 7 6. x + 3z = 9 7. x + 7z = 4 x 4 + z = 6 x + 3 = 4 x + + z = 3 x + 4 + z = x 5 + 5z = 7 3x + 9 36z = 33 Solve the sstem b Gaussian elimination. (Section.) 8. x + 4z = 3 9. 3x 3 + 6z = 6. x + z = 5 x + 4 z = 7 x + z = 5 3x z = 5x 3 + z = 3 5x 8 + 3z = 7 6x + 5z = 3. Contestants participate in a pumpkin carving contest. The table shows the results of the voting for the gold, silver, and bronze medalists. The gold medalist earned 38 points, the silver medalist earned 3 points, and the bronze medalist earned points. How man points is each vote worth? (Sections. and.) Medal st place nd place 3rd place gold 6 silver 3 4 3 bronze 4 5. In a football game, a total of 45 points were scored. During the game, there were 3 scoring plas. These plas were a combination of touchdowns, extra-point kicks, and field goals, which are worth 6 points, point, and 3 points, respectivel. The same number of extra-point kicks and touchdowns were scored, and there were six times as man touchdowns as field goals. How man touchdowns, extra-points, and field goals were scored during the game? (Sections. and.) 3. A small corporation borrowed $8, to expand its business. Some of the mone was borrowed at each 8%, some at 9%, and some at %. The simple interest charged was $67,5 and the amount borrowed at 8% was four times the amount borrowed at %. How much mone was borrowed at each rate? (Sections. and.) 74 Chapter Solving Sstems of Equations and Inequalities
.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.3.B Solving Linear Sstems Using Technolog Essential Question How can ou represent algebraic expressions using a coefficient matrix? A matrix is a rectangular arrangement of numbers. The dimensions of a matrix with m rows and n columns are m n. So, the dimensions of matrix A are 3. A = [ 4 6 5 3 ] rows 3 columns Writing Coefficient Matrices Work with a partner. Match each set of algebraic expressions with its coefficient matrix. Explain our reasoning. Sample Algebraic x + + 6z Coefficient expressions: x + 3z matrix: [ a. 4x + 3 b. 4x + 3z 5x + 5x + c. 4x 3z d. 4x + +3z 5x 3 z 4z 6 3 ] A. [ 4 5 C. [ 4 5 3 B. ] [ 4 3 ] D. [ 4 5 3 ] 3 4 3 ] Writing Coefficient Matrices ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, ou need to analze mathematical relationships to connect mathematical ideas. Work with a partner. Write and enter the coefficient matrix for each set of expressions into a graphing calculator. a. 5z b. 5 z c. x 3 x + x + 4z x + z x + + 9z Communicate Your Answer 3. How can ou represent algebraic expressions using a coefficient matrix? 4. Write the algebraic expressions that are represented b the coefficient matrix. MATRIX[A] 4 4 [ - 4 ] [3 5-8] [4 3 ] [-9 ] Section.3 Solving Linear Sstems Using Technolog 75
.3 Lesson What You Will Learn Core Vocabular matrix, p. 76 dimensions of a matrix, p. 76 elements of a matrix, p. 76 augmented matrix, p. 76 Previous sstem of three linear equations row-echelon form Write augmented matrices for sstems of linear equations. Use technolog to solve sstems of linear equations in three variables. Writing Augmented Matrices for Sstems A matrix is a rectangular arrangement of numbers. The dimensions of a matrix with m rows and n columns are m n (read m b n ). So, the dimensions of matrix A are 3. The numbers in the matrix are its elements. A = [ 4 6 3 columns 5 3 ] rows The element in the first row and third column is 5. A matrix derived from a sstem of linear equations (each written in standard form with the constant term on the right) is the augmented matrix of the sstem. For example, the sstem below can be represented b the given augmented matrix. [ Sstem: 5x + 3 z = Augmented x + + 3z = matrix: 3x 4 + z = 5 3 coefficients 3 4 3......... ] constants Before ou write an augmented matrix, make sure each equation in the sstem is written in standard form. Include zeros for the coefficients of an missing variables. This determines the order of the constants and coefficients in the augmented matrix. Writing an Augmented Matrix Write an augmented matrix for the sstem. Then state the dimensions. SOLUTION x = x + 3 = Begin b rewriting each equation in the sstem in standard form. x + = x + 3 = Next, use the coefficients and constants as elements of the augmented matrix. [ 3 ] The augmented matrix has two rows and three columns, so the dimensions are 3....... 76 Chapter Solving Sstems of Equations and Inequalities
Writing an Augmented Matrix COMMON ERROR Because the second equation does not have a z-term, the coefficient of z is. Write an augmented matrix for the sstem. Then state the dimensions. 3x 4 = 7 z 9x + 3 = 3 x + 4 z = SOLUTION Begin b rewriting each equation in the sstem in standard form. 3x 4 + z = 7 9x + 3 + z = 3 x + 4 z = Next, use the coefficients and constants as elements of the augmented matrix. [ 3 4 7 9 3 3 4......... ] The augmented matrix has three rows and four columns, so the dimensions are 3 4. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write an augmented matrix for the sstem. Then state the dimensions.. x 8 = 4. x + = z 3. 9x 8 + 3z = 5x + = 9 7x + 9 z = x + z = 6 6x + 4 + 8z = x + 4 = 6 Solving Sstems of Equations Using Technolog Man technolog tools have matrix features that ou can use to solve a sstem of linear equations. The augmented matrix in Example, rewritten in reduced row-echelon form, is shown below. Observe that the solution to this sstem is (x,, z) = (.4, 5., 6). You can verif this solution in the original sstem. [..........4 5. 6 ] Core Concept Solving a Linear Sstem Using Technolog Step Write an augmented matrix for the linear sstem. Step Enter the augmented matrix into our graphing calculator. Step 3 Use the reduced row-echelon form feature to rewrite the sstem. Step 4 Interpret the result from Step 3 to solve the linear sstem. Section.3 Solving Linear Sstems Using Technolog 77
Solving a Sstem Using Technolog Use a graphing calculator to solve the sstem. x + + z = x + 3 + z = 3 x z = 6 REMEMBER An m n matrix has m rows and n columns. SOLUTION Step Write an augmented matrix for the linear sstem. [ 3 3 6 Step Enter the dimensions and elements of the augmented matrix into our graphing calculator.......... ] NAMES :[A] :[B] 3:[C] 4:[D] 5:[E] 6:[F] 7:[G] MATH EDIT MATRIX[A] 3 4 [ ] [ 3 3 ] [ - - -6] Step 3 Use the reduced row-echelon form feature to rewrite the sstem. NAMES MATH 6:randM( 7:augment 8:Matr list( 9:List matr( :cumsum A:ref( B:rref( EDIT rref([a] [ [ -] [ ] [ ]] Step 4 Converting the matrix back to a sstem of linear equations, ou have: x = = z = The solution is x =, =, and z =, or the ordered triple (,, ). Check this solution in each of the original equations. Check x + + z = x + 3 + z = 3 x z = 6 + + =? ( ) + 3() + =? 3 () =? 6 = 3 = 3 6 = 6 Monitoring Progress Use a graphing calculator to solve the sstem. Help in English and Spanish at BigIdeasMath.com 4. x + 3z = 5. x + 3 z = 6. 3x + 5z = x + z = x 6 + z = 3 6x z = 8 3x + 4 4z = 4 3x + 5 z = 4 + 3z = 78 Chapter Solving Sstems of Equations and Inequalities
.3 Exercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETING THE SENTENCE A matrix derived from a sstem of linear equations is called the matrix of the sstem.. WRITING Describe how to find the solution of a sstem of linear equations in three variables using technolog. Monitoring Progress and Modeling with Mathematics In Exercises 3, write an augmented matrix for the sstem. Then state the dimensions. (See Examples and.) 3. 3x 4 = 7 4. 5 4x = 7 9x + = 3x 7 = 5 5. x + 8 7z = 6. 4x 5 + z = 3 5x + 9 + 5z = 5 6x + 4 + 9z = 8 6z 3 8x = x z = 7 7. x + z = 4 8. 5x + z = 9 6x 5z = 3 3x + 5 8z = 5 3x + 7 + 8z = 5 4x + + 9z = 9. 3x + = z + 7. x + 3 = 5z + 9 5x + 4z = 8 x 4 + 5z = 3 x + 9 3z = 6 4 + 3z = 6x ERROR ANALYSIS In Exercises and, describe and correct the error in writing an augmented matrix for the sstem below... 3x 9 + 5z = 8 x + = 5 6x 9z + 4 = 4 The augmented matrix is: [ 3 9 5 8 5 6 9 4 4......... The augmented matrix is: [ 3 9 5 8 5 6 4 9 4......... ] ] In Exercises 3, use a graphing calculator to solve the sstem. (See Example 3.) 3. 4x + + 6z = 7 4. x + 4 z = 7 3x + 3 + z = 7 x + z = 5 x + z = 9 3x + 3z= 5. x + + z = 9 6. x + 3z = 9 x + + z = 3 x + 5 + z = 5x z = 3x 6 + 9z = 7. x + 3z = 6 8. 4x + + 6z = x + 3 + z = x + + 4z = 3x + z = 3 x + z = 5 9. x + + 4z = 7. x + + z = x 3 z = 4 x + z = 4x + + z = 8 3x + 3 + 6z = 4. x + + z =. x + 4 + z = x + + z = 5 3x 3 z = x + 4 + 3z = 5 5x z = 8 3. MODELING WITH MATHEMATICS A compan sells three tpes of gift baskets. The basic basket has two movie passes and one package of microwave popcorn, and costs $5.5. The medium basket has two movie passes, two packages of microwave popcorn, and one DVD, and costs $37. The super basket has four movie passes, three packages of microwave popcorn, and two DVDs, and costs $7.5. a. Write an augmented matrix to represent the situation. b. Use a graphing calculator to find the cost of each basket item. Section.3 Solving Linear Sstems Using Technolog 79
4. MODELING WITH MATHEMATICS You go shopping at a local department store with our friend and cousin. You bu one pair of jeans, four pairs of shorts, and two shirts for $84. Your friend bus two pairs of jeans, one pair of shorts, and three shirts for $76. Your cousin bus one pair of jeans, two pairs of shorts, and one shirt for $5. a. Write an augmented matrix to represent the situation. b. Use a graphing calculator to find the cost of each piece of clothing. 5. MODELING WITH MATHEMATICS You have 85 coins in nickels, dimes, and quarters with a combined value of $3.5. There are twice as man quarters as dimes. a. Write an augmented matrix to represent the situation. b. Use a graphing calculator to find the number of each tpe of coin. 6. HOW DO YOU SEE IT? Write a sstem of equations for the augmented matrix below. [ 3 4 8 9 4 6......... 7. MAKING AN ARGUMENT Your friend states that the number of rows in an augmented matrix of the sstem will alwas be the same as the number of variables in the sstem. Is our friend correct? Explain our reasoning. ] 8. USING TOOLS Use a graphing calculator to solve the sstem of four linear equations in four variables. w + 5x 4 + 6z = x + 7z = 5 4w + 8x 7 + 4z = 5 3w + 6x 5 + z = 6 9. REASONING Is it possible to write more than one augmented matrix for a sstem of linear equations? Explain our reasoning. 3. MATHEMATICAL CONNECTIONS The sum of the measures of the angles in ABC is 8. The sum of the measures of angle B and angle C is twice the measure of angle A. The measure of angle B is 3 less than the measure of angle C. a. Write an augmented matrix to represent the situation. b. Use a graphing calculator to find the measures of the three angles. 3. ABSTRACT REASONING Let a, b, and c be real numbers. Classif the linear sstem represented b each matrix as consistent or inconsistent. Explain our reasoning. a. b. c. [ [ [........................... ] a b c a b ] ] a b 3. THOUGHT PROVOKING Write an augmented matrix, not in reduced row-echelon form, for a sstem that has exactl one solution, (x,, z) = (, 3, 5). Justif our answer. Maintaining Mathematical Proficienc Solve the inequalit. Graph the solution. (Skills Review Handbook) 33. x > 5 34. 5z + 8 7 35. w + 7 3w + 5 36. r + < 3 r + 7 Reviewing what ou learned in previous grades and lessons Graph the inequalit in a coordinate plane. (Skills Review Handbook) 37. < 3x 38. 4 > x + 6 39. x + 5 4. x 3 9 8 Chapter Solving Sstems of Equations and Inequalities
.4 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.3.E A.3.F A.3.G Solving Sstems of Linear Inequalities Essential Question How can ou graph a sstem of three linear inequalities? Graphing Linear Inequalities Work with a partner. Match each linear inequalit with its graph. Explain our reasoning. x + 4 Inequalit x Inequalit 3 Inequalit 3 A. B. 4 C. 4 x 4 4 x 4 x USING PRECISE MATHEMATICAL LANGUAGE To be proficient in math, ou need to explain mathematical ideas using precise mathematical language. Graphing a Sstem of Linear Inequalities Work with a partner. Consider the linear inequalities given in Exploration. x + 4 Inequalit x Inequalit 3 Inequalit 3 a. Use three different colors to graph the inequalities in the same coordinate plane. What is the result? b. Describe each of the shaded regions of the graph. What does the unshaded region represent? Communicate Your Answer 3. How can ou graph a sstem of three linear inequalities? 4. When graphing a sstem of three linear inequalities, which region represents the solution of the sstem? 5. Do ou think all sstems of three linear inequalities have a solution? Explain our reasoning. 6. Write a sstem of three linear inequalities represented b the graph. 3 4 x 3 Section.4 Solving Sstems of Linear Inequalities 8
.4 Lesson What You Will Learn Core Vocabular sstem of linear inequalities, p. 8 solution of a sstem of linear inequalities, p. 8 graph of a sstem of linear inequalities, p. 83 Previous linear inequalit in two variables Check solutions of sstems of linear inequalities. Graph sstems of linear inequalities. Write sstems of linear inequalities. Use sstems of linear inequalities to solve real-life problems. Sstems of Linear Inequalities A sstem of linear inequalities is a set of two x + Inequalit or more linear inequalities in the same variables. < x + Inequalit An example is shown. < x Inequalit 3 A solution of a sstem of linear inequalities is an ordered pair that is a solution of each inequalit in the sstem. Checking Solutions Tell whether each ordered pair is a solution of the sstem of linear inequalities. < 4x Inequalit x Inequalit x + Inequalit 3 a. (, ) b. (, 8) SOLUTION a. Substitute for x and for in each inequalit. Inequalit Inequalit Inequalit 3 < 4x x x + <? 4()?? () + < 4 3 Because the ordered pair (, ) is a solution of each inequalit, it is a solution of the sstem. b. Substitute for x and 8 for in each inequalit. Inequalit Inequalit Inequalit 3 < 4x x x + 8 <? ( ) 8? 8? ( ) + 8 < 4 8 3 8 3 Because (, 8) is not a solution of each inequalit, it is not a solution of the sstem. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the ordered pair is a solution of the sstem of linear inequalities. < 3. (, ); > x + 3 x 4. (, 3); 3x + x 5 < x + 6 8 Chapter Solving Sstems of Equations and Inequalities
Graphing Sstems of Linear Inequalities The graph of a sstem of linear inequalities is the graph of all the solutions of the sstem. Core Concept Graphing a Sstem of Linear Inequalities Step Graph each inequalit in the same coordinate plane. Step Find the intersection of the x half-planes that are solutions of the inequalities. This intersection 4 is the graph of the sstem. 3 x + < x + x Check Graphing a Sstem of Linear Inequalities Verif that (, 3) is a solution of each inequalit. Inequalit Inequalit Inequalit 3 > x 5 3 >? () 5 3 > 9 x + 3 3? + 3 3 5 Graph the sstem of linear inequalities. > x 5 Inequalit x + 3 Inequalit x Inequalit 3 SOLUTION Step Graph each inequalit. Step Find the intersection of the half-planes. One solution is (, 3). 5 (, 3) x x 3? 3 Graphing a Sstem of Linear Inequalities (No Solution) Graph the sstem of linear inequalities. x + 3 < 6 Inequalit 3 x + 4 Inequalit x + < Inequalit 3 SOLUTION Step Graph each inequalit. 5 3 Step Find the intersection of the half-planes. Notice that there is no region shaded red, blue, and green. 5 x So, the sstem has no solution. Section.4 Solving Sstems of Linear Inequalities 83
Monitoring Progress Graph the sstem of linear inequalities. Help in English and Spanish at BigIdeasMath.com 3. < 3x 4. x + > 3 5. x 4 > x + 4 6x + < 4x 5 x + 8 x + Writing Sstems of Linear Inequalities Writing a Sstem of Linear Inequalities Write a sstem of linear inequalities represented b the graph. SOLUTION Inequalit The vertical boundar line passes through ( 3, ). So, an equation of the line is x = 3. The shaded region is to the right of the solid boundar line, so the inequalit is x 3. 4 4 4 4 x Inequalit The slope of the boundar line is, and the -intercept is. So, an equation of the line is = x. The shaded region is above the dashed boundar line, so the inequalit is > x. Inequalit 3 The slope of the boundar line is, and the -intercept is. So, an equation of the line is = x +. The shaded region is below the dashed boundar line, so the inequalit is < x +. The sstem of inequalities represented b the graph is x 3 Inequalit > x Inequalit < x +. Inequalit 3 Monitoring Progress Write a sstem of inequalities represented b the graph. Help in English and Spanish at BigIdeasMath.com 6. 4 7. 4 4 4 x 4 4 x 4 84 Chapter Solving Sstems of Equations and Inequalities
Solving Real-Life Problems Appling Mathematics A discount shoe store is having a sale, as described in the advertisement shown. Use the information in the ad to write and graph a sstem of inequalities that represents the regular and possible sale prices. How much can regularl priced shoes cost on sale? SOLUTION. Understand the Problem You know the range of regular prices and the range of discounts. You are asked to write and graph a sstem that represents the situation and determine how much regularl priced shoes cost on sale.. Make a Plan Use the given information to write a sstem of inequalities. Then graph the sstem and identif an ordered pair in the solution region. 3. Solve the Problem Write a sstem of inequalities. Let x be the regular price and let be the sale price. x Regular price must be at least $. x 8 Regular price must be at most $8..4x.9x Sale price is at least ( 6)% = 4% of regular price. Sale price is at most ( )% = 9% of regular price. Graph the sstem. Check x 5 x 8 5 8.4x 3?.4(5) 3 Sale price (dollars) Footwear Sale 8 7 6 5 4 3 3 4 5 6 7 8 x Regular price (dollars).9x 3?.9(5) 3 45 From the graph, ou can see that one ordered pair in the solution region is (5, 3). So, a $5 pair of shoes could cost $3 on sale. 4. Look Back Check our solution b substituting x = 5 and = 3 in each inequalit in the sstem, as shown. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. Identif and interpret another solution of Example 5. 9. WHAT IF? Suppose all the shoes were sold except those regularl priced from $6 to $8. How does this change the sstem? Is (5, 3) still a solution? Explain. Section.4 Solving Sstems of Linear Inequalities 85
.4 Exercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY What must be true in order for an ordered pair to be a solution of a sstem of linear inequalities?. WHICH ONE DOESN T BELONG? Use the graph shown. Which ordered pair does not belong with the other three? Explain our reasoning. ( 3, ) ( 3, ) (, ) (, 4) 5 3 x Monitoring Progress and Modeling with Mathematics In Exercises 3 8, tell whether the ordered pair is a solution of the sstem of linear inequalities. (See Example.) 3. (, ); 5. (, ); 5x 6 3x + > x + < 3x + x < 4 > x 4. (8, ); 6. (, 5); < x + 5 4x < x x + 3 x 3x > 6 5. 6. < 5 x 9 > 6 x + < 5 x + < x x + 3 In Exercises 7, write a sstem of linear inequalities represented b the graph. (See Example 4.) 7. 3 8. 7. (, ); x + 4 x + > x x 8. (, 3); x + < 5 x + 3 3x > 4 3x 3 In Exercises 9 6, graph the sstem of linear inequalities. (See Examples and 3.) 9. 3 x 3. 3 3 x 9. < 4. x > 3 x 6 > x < x 5 3 x 3 x. < x + 4. < x + 4x 3 < 3x < x 3 > 3x +. 3. 4 3. x 3 > 6 4. x 4 > 5x 3 < 3 x + x + 3 > 3 x + 3 > 4 x x 86 Chapter Solving Sstems of Equations and Inequalities