1.4 Solving Linear Sstems 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. Eplain our reasoning. Then classif the sstem as consistent or inconsistent. a. 3 = 3 b. 3 = 3 c. 3 = 3 4 + 6 = 6 + = 5 4 + 6 = 6 A. B. C. 4 4 4 Solving Sstems of Linear Equations Work with a partner. Solve each linear sstem b substitution or elimination. Then use the graph of the sstem below to check our solution. a. + = 5 b. + 3 = 1 c. + = 0 = 1 + = 4 3 + = 1 4 FINDING AN ENTRY POINT To be proficient in math, ou need to look for entr points to the solution of a problem. 4 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. Eplain how ou would approach solving such a sstem. 5. Appl our strateg in Question 4 to solve the linear sstem. + + z = 1 Equation 1 z = 3 Equation + z = 1 Equation 3 Section 1.4 Solving Linear Sstems 9
1.4 Lesson Core Vocabular linear equation in three variables, p. 30 sstem of three linear equations, p. 30 solution of a sstem of three linear equations, p. 30 ordered triple, p. 30 Previous sstem of two linear equations What You Will Learn Visualize solutions of sstems of linear equations in three variables. Solve sstems of linear equations in three variables algebraicall. Solve real-life problems. Visualizing Solutions of Sstems A linear equation in three variables,, and z is an equation of the form a + b + cz = d, where a, b, and c are not all zero. The following is an eample of a sstem of three linear equations in three variables. 3 + 4 8z = 3 Equation 1 + + 5z = 1 Equation 4 + z = 10 Equation 3 A solution of such a sstem is an ordered triple (,, 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. Eactl 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. 30 Chapter 1 Linear Functions
Solving Sstems of Equations Algebraicall The algebraic methods ou used to solve sstems of linear equations in two variables can be etended to solve a sstem of linear equations in three variables. LOOKING FOR STRUCTURE The coefficient of 1 in Equation 3 makes a convenient variable to eliminate. ANOTHER WAY In Step 1, ou could also eliminate to get two equations in and z, or ou could eliminate z to get two equations in and. Core Concept Solving a Three-Variable Sstem Step 1 Rewrite the linear sstem in three variables as a linear sstem in two variables b using the substitution or elimination method. 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 0 = 1, in an of the steps, the sstem has no solution. When ou do not obtain a false equation, but obtain an identit such as 0 = 0, the sstem has infinitel man solutions. Solving a Three-Variable Sstem (One Solution) Solve the sstem. 4 + + 3z = 1 Equation 1 SOLUTION 3 + 5z = 7 Equation 6 + 4z = 3 Equation 3 Step 1 Rewrite the sstem as a linear sstem in two variables. 4 + + 3z = 1 Add times Equation 3 to 1 + 8z = 6 Equation 1 (to eliminate ). 16 + 11z = 6 New Equation 1 3 + 5z = 7 18 + 3 1z = 9 Add 3 times Equation 3 to Equation (to eliminate ). 16 7z = New Equation Step Solve the new linear sstem for both of its variables. 16 + 11z = 6 Add new Equation 1 16 7z = and new Equation. 4z = 8 z = Solve for z. = 1 Substitute into new Equation 1 or to find. Step 3 Substitute = 1 and z = into an original equation and solve for. 6 + 4z = 3 Write original Equation 3. 6( 1) + 4() = 3 Substitute 1 for and for z. = 5 Solve for. The solution is = 1, = 5, and z =, or the ordered triple ( 1, 5, ). Check this solution in each of the original equations. Section 1.4 Solving Linear Sstems 31
Solving a Three-Variable Sstem (No Solution) Solve the sstem. + + z = Equation 1 5 + 5 + 5z = 3 Equation 4 + 3z = 6 Equation 3 SOLUTION Step 1 Rewrite the sstem as a linear sstem in two variables. 5 5 5z = 10 Add 5 times Equation 1 5 + 5 + 5z = 3 to Equation. 0 = 7 Because ou obtain a false equation, the original sstem has no solution. ANOTHER WAY Subtracting Equation from Equation 1 gives z = 0. After substituting 0 for z in each equation, ou can see that each is equivalent to = + 3. Solving a Three-Variable Sstem (Man Solutions) Solve the sstem. + z = 3 Equation 1 SOLUTION z = 3 Equation 5 5 + z = 15 Equation 3 Step 1 Rewrite the sstem as a linear sstem in two variables. + z = 3 z = 3 Add Equation 1 to Equation (to eliminate z). = 6 New Equation z = 3 5 5 + z = 15 Add Equation to Equation 3 (to eliminate z). 6 6 = 18 New Equation 3 Step Solve the new linear sstem for both of its variables. 6 + 6 = 18 Add 3 times new Equation 6 6 = 18 to new Equation 3. 0 = 0 Because ou obtain the identit 0 = 0, 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 = + 3. Then substitute + 3 for in original Equation 1 to obtain z = 0. So, an ordered triple of the form (, + 3, 0) is a solution of the sstem. 3 Chapter 1 Linear Functions Monitoring Progress Solve the sstem. Check our solution, if possible. Help in English and Spanish at BigIdeasMath.com 1. + z = 11. + z = 1 3. + + z = 8 3 + z = 7 4 + 4 4z = + z = 8 + + 4z = 9 3 + + z = 0 + + z = 16 4. In Eample 3, describe the solutions of the sstem using an ordered triple in terms of.
Solving Real-Life Problems Solving a Multi-Step Problem 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 $30 for each lawn seat. There are three times as man seats in Section B as in Section A. The revenue from selling all 3,000 seats is $870,000. How man seats are in each section of the amphitheater? STAGE SOLUTION Step 1 Write a verbal model for the situation. seats in B, = 3 seats in A, seats in A, + seats in B, + lawn seats, z = Total number of seats 75 seats in A, + 55 seats in B, + 30 lawn seats, z = Total revenue Step Write a sstem of equations. = 3 Equation 1 + + z = 3,000 Equation 75 + 55 + 30z = 870,000 Equation 3 Step 3 Rewrite the sstem in Step as a linear sstem in two variables b substituting 3 for in Equations and 3. + + z = 3,000 Write Equation. + 3 + z = 3,000 Substitute 3 for. 4 + z = 3,000 New Equation 75 + 55 + 30z = 870,000 Write Equation 3. 75 + 55(3) + 30z = 870,000 Substitute 3 for. 40 + 30z = 870,000 New Equation 3 STUDY TIP When substituting to find values of other variables, choose original or new equations that are easiest to use. Step 4 Solve the new linear sstem for both of its variables. 10 30z = 690,000 Add 30 times new Equation 40 + 30z = 870,000 to new Equation 3. 10 = 180,000 = 1500 Solve for. = 4500 Substitute into Equation 1 to find. z = 17,000 Substitute into Equation to find z. The solution is = 1500, = 4500, and z = 17,000, or (1500, 4500, 17,000). So, there are 1500 seats in Section A, 4500 seats in Section B, and 17,000 lawn seats. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. WHAT IF? On the first da, 10,000 tickets sold, generating $356,000 in revenue. The number of seats sold in Sections A and B are the same. How man lawn seats are still available? Section 1.4 Solving Linear Sstems 33
1.4 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY The solution of a sstem of three linear equations is epressed as a(n).. WRITING Eplain how ou know when a linear sstem in three variables has infinitel man solutions. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, solve the sstem using the elimination method. (See Eample 1.) 3. + z = 5 4. + 4 6z = 1 + + z = + z = 7 + 3 z = 9 + 4z = 5 5. + z = 9 6. 3 + z = 8 + 6 + z = 17 3 + 4 + 5z = 14 5 + 7 + z = 4 3 + 4z = 14 7. + + 5z = 1 8. 3 + 3z = + z = 7 + 5z = 14 + 4 3z = 14 + 4 + z = 6 13. + 3 z = 14. + z = 3 + z = 0 + z = 1 3 + 3z = 1 6 3 z = 7 15. + + 3z = 4 16. 3 + z = 6 3 + z = 1 + z = 5 4z = 14 7 + 8 6z = 31 17. MODELING WITH MATHEMATICS Three orders are placed at a pizza shop. Two small pizzas, a liter of soda, and a salad cost $14; one small pizza, a liter of soda, and three salads cost $15; and three small pizzas, a liter of soda, and two salads cost $. How much does each item cost? ERROR ANALYSIS In Eercises 9 and 10, describe and correct the error in the first step of solving the sstem of linear equations. 4 + z = 18 + + z = 11 3 + 3 4z = 44 9. 10. 4 + z = 18 4 + + z = 11 + 3z = 7 1 3 + 6z = 18 3 + 3 4z = 44 15 + z = 6 18. 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? Eplain. SAM S Furniture Store Sofa and love seat In Eercises 11 16, solve the sstem using the elimination method. (See Eamples and 3.) 11. 3 + z = 4 1. 5 + z = 6 6 + 4z = 8 + + z = + 3z = 10 1 + 4 = 10 Sofa and two chairs Sofa, love seat, and one chair 34 Chapter 1 Linear Functions
In Eercises 19 8, solve the sstem of linear equations using the substitution method. (See Eample 4.) 19. + + 6z = 1 0. 6 z = 8 3 + + 5z = 16 + 5 + 3z = 7 + 3 4z = 11 3 4z = 18 1. + + z = 4. + = 1 5 + 5 + 5z = 1 + 3 + z = 4 4 + z = 9 + 4z = 10 3. 3 + z = 10 4. = 4 + z = 13 + = 6 z = 5 4 3 + z = 6 5. + z = 4 6. z = 15 3 + + 4z = 17 4 + 5 + z = 10 + 5 + z = 8 4 + 3z = 0 7. 4 + + 5z = 5 8. + z = 3 8 + + 10z = 10 + 4 z = 6 z = + z = 6 9. PROBLEM SOLVING The number of left-handed people in the world is one-tenth the number of righthanded people. The percent of right-handed people is nine times the percent of left-handed people and ambidetrous people combined. What percent of people are ambidetrous? 31. WRITING Eplain when it might be more convenient to use the elimination method than the substitution method to solve a linear sstem. Give an eample to support our claim. 3. REPEATED REASONING Using what ou know about solving linear sstems in two and three variables, plan a strateg for how ou would solve a sstem that has four linear equations in four variables. MATHEMATICAL CONNECTIONS In Eercises 33 and 34, write and use a linear sstem to answer the question. 33. The triangle has a perimeter of 65 feet. What are the lengths of sides, m, and n? = 1 m 3 m n = + m 15 34. What are the measures of angles A, B, and C? A (5A C) A B (A + B) C 30. 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 0 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 1 point. Lawrence had a strong second-place showing, with as man second place finishers as first- and third-place finishers combined. 35. OPEN-ENDED Consider the sstem of linear equations below. Choose nonzero values for a, b, and c so the sstem satisfies the given condition. Eplain our reasoning. + + z = a + b + cz = 10 + z = 4 a. The sstem has no solution. b. The sstem has eactl one solution. c. The sstem has infinitel man solutions. 36. 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? Eplain our reasoning. Section 1.4 Solving Linear Sstems 35
37. PROBLEM SOLVING A contractor is hired to build an 40. HOW DO YOU SEE IT? Determine whether the apartment comple. Each 840-square-foot unit has a bedroom, kitchen, and bathroom. The bedroom will be the same size as the kitchen. The owner orders 980 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. Eplain our reasoning. a. b. KITCHEN 41. CRITICAL THINKING Find the values of a, b, and c so that the linear sstem shown has ( 1,, 3) as its onl solution. Eplain our reasoning. BEDROOM Total Area: 840 ft + 3z = a + z = b + 3 z = c 38. THOUGHT PROVOKING Does the sstem of linear equations have more than one solution? Justif our answer. 4. ANALYZING RELATIONSHIPS Determine which arrangement(s) of the integers 5,, and 3 produce a solution of the linear sstem that consist of onl integers. Justif our answer. 4 + + z = 0 + 1 3z = 0 14 z = 0 3 + 6z = 1 + + z = 30 5 + z = 6 39. PROBLEM SOLVING A florist must make 5 identical bridesmaid bouquets for a wedding. The budget is $160, and each bouquet must have 1 flowers. Roses cost $.50 each, lilies cost $4 each, and irises cost $ each. The florist wants twice as man roses as the other two tpes of flowers combined. 43. 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 maimum budget. 0 10 0 30 40 50 60 70 80 90 100 110 0 10 130 140 150 160 170 180 190 00 0 10 0 30 40 50 60 70 80 90 100 110 0 10 130 140 150 160 170 180 190 00 0 10 0 30 40 50 60 70 80 90 100 110 10 130 140 150 160 170 180 190 00 b. Solve the sstem to find how man of each tpe of flower should be in each bouquet. c. Suppose there is no limitation on the total cost of the bouquets. Does the problem still have eactl one solution? If so, find the solution. If not, give three possible solutions. 0 10 0 30 40 50 60 70 80 90 100 110 10 130 140 150 160 170 180 190 00 Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Simplif. (Skills Review Handbook) 44. ( ) 45. (3m + 1) 46. (z 5) Write a function g described b the given transformation of f() = 5. 36 47. (4 ) (Section 1.) 48. translation units to the left 49. reflection in the -ais 50. translation 4 units up 51. vertical stretch b a factor of 3 Chapter 1 hsnb_alg_pe_0104.indd 36 Linear Functions /5/15 9:57 AM