Essential Question How can you determine the number of solutions of a linear system?
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1 .1 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. Eplain our reasoning. Then classif the sstem as consistent or inconsistent. a. 3 = 3 b. 3 = 3 c. 3 = = 6 + = = 6 A. B. C 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. + = 5 b. + 3 = 1 c. + = 0 = 1 + = = 1 4 FORMULATING A PLAN To be proficient in math, ou need to formulate a plan to solve 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 solve such a sstem b substitution. 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 Solving Linear Sstems Using Substitution 59
2 .1 Lesson What You Will Learn Core Vocabular linear equation in three variables, p. 60 sstem of three linear equations, p. 60 solution of a sstem of three linear equations, p. 60 ordered triple, p. 60 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,, 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 z = 3 Equation z = 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. 60 Chapter Solving Sstems of Equations and Inequalities
3 Solving Sstems of Equations b Substitution The substitution method for solving sstems of linear equations in two variables can be etended to solve a sstem of linear equations in three variables. Core Concept Solving a Three-Variable Sstem b Substitution Step 1 Solve one equation for one of its variables. Step Substitute the epression from Step 1 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 -term in Equation 1 makes it convenient to solve for or z. ANOTHER WAY In Step 1, ou could also solve Equation 1 for z. 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 b substitution. 3 6z = 6 Equation 1 SOLUTION + 4z = 10 Equation + z = 1 Equation 3 Step 1 Solve Equation 1 for. = z New Equation 1 Step Substitute z for in Equations and 3 to obtain a sstem in two variables. (z ) + 4z = 10 Substitute z for in Equation. + z = 8 New Equation + (z ) z = 1 Substitute z for in Equation z = 16 New Equation 3 Step 3 Solve the new linear sstem for both of its variables. = 8 z Solve new Equation for. (8 z) + 3z = 16 Substitute 8 z for in new Equation 3. z = 0 Solve for z. = 8 Substitute into new Equation 3 to find. Step 4 Substitute = 8 and z = 0 into an original equation and solve for. 3 6z = 6 Write original Equation (0) = 6 Substitute 0 for z. = Solve for. The solution is = 8, =, and z = 0, or the ordered triple (8,, 0). Check this solution in each of the original equations. Section.1 Solving Linear Sstems Using Substitution 61
4 Solving a Three-Variable Sstem (No Solution) Solve the sstem b substitution. 4 + z = 5 Equation 1 SOLUTION 8 + z = 1 Equation + + 7z = 3 Equation 3 Step 1 Solve Equation 1 for z. z = New Equation 1 Step Substitute for z in Equations and 3 to obtain a sstem in two variables. 8 + ( ) = 1 Substitute for z in Equation. 10 = 1 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. 4 + z = Equation 1 SOLUTION z = Equation z = 6 Equation 3 Step 1 Solve Equation 1 for. = 4 + z + New Equation 1 Step Substitute 4 + z + for in Equations and 3 to obtain a sstem in two variables. 4 + ( 4 + z + ) + z = Substitute 4 + z + for in Equation. z = 0 New Equation 1 + 3( 4 + z + ) 3z = 6 Substitute 4 + 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 0 for z in Equation 1 to obtain = 4 +. So, an ordered triple of the form (, 4 +, 0) 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 z = 7. + z = z = z = 5 z = 4 + 1z = z = z = z = 1 4. In Eample 3, describe the solutions of the sstem using an ordered triple in terms of. 6 Chapter Solving Sstems of Equations and Inequalities
5 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 $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 z = 3,000 Equation z = 870,000 Equation 3 Step 3 Substitute 3 for in Equations and 3 to obtain a sstem in two variables z = 3,000 Substitute 3 for in Equation. 4 + z = 3,000 New Equation (3) + 30z = 870,000 Substitute 3 for in Equation z = 870,000 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 ( 4 + 3,000) = 870,000 z = 4 + 3,000 Solve new Equation for z. = 1500 Solve for. Substitute 4 + 3,000 for z in new Equation 3. = 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 Solving Linear Sstems Using Substitution 63
6 .1 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).. DIFFERENT WORDS, SAME QUESTION Consider the sstem of linear equations shown. Which is different? Find both answers. Solve the sstem of linear equations. + 3 = z = 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 Eercises 3 and 4, determine whether the ordered triple is a solution of the sstem. Justif our answer. 3. (4, 5, 1) 4. (, 3, 6) + + 5z = z = z = z = 13 + z = z = 3 In Eercises 5 14, solve the sstem b substitution. (See Eample 1.) 5. = z = 10 + = 6 + z = z = 6 z = = z = z = 4 5 z = z = z = z = z = z = z = 0 + = z = z = z = z = z = 37 + z = 0 + 3z = z = z = z = z = z = z = 6 ERROR ANALYSIS In Eercises 15 and 16, describe and correct the error in the first steps of solving the sstem of linear equations. + z = z = 11 z = z = = 3 = 9 = + z + ( + z) z = 3 z = 5 In Eercises 17, solve the sstem b substitution. (See Eamples and 3.) z = = z + + z = z = z = z = z = z = z = z = z = 4 + z = z = z = z = z = z = z = Chapter Solving Sstems of Equations and Inequalities
7 3. MODELING WITH MATHEMATICS A wholesale store advertises that for $0 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, 1 pound of cashews, and 3 pounds of almonds for $36. What is the price per pound of each tpe of nut? (See Eample 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 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. 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 1st place nd place 3rd place Points Plaer Plaer Plaer 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. Eplain 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 ambidetrous people combined. What percent of people are ambidetrous? MATHEMATICAL CONNECTIONS In Eercises 9 and 30, 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? = 1 m 3 m n = + m What are the measures of angles A, B, and C? B A (5A C) A (A + B) C 31. OPEN-ENDED Write a sstem of three linear equations in three variables that has the ordered triple ( 4, 1, ) 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? Eplain our reasoning. Section.1 Solving Linear Sstems Using Substitution 65
8 33. PROBLEM SOLVING A contractor is hired to build an 36. 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 37. REASONING Consider a sstem of three linear equations in three variables. Describe the possible number of solutions in each situation. BEDROOM Total Area: 840 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. 3 + z = 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, 0, and 1 to write a linear sstem that has a solution of (30, 0, 17). b. Make a conjecture about the minimum number of equations that a linear sstem in n variables can have when there is eactl one solution z = z = z = 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. 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 maimum budget. 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. Reviewing what ou learned in previous grades and lessons Solve the sstem of linear equations b elimination. (Skills Review Handbook) = = = = = Chapter Maintaining Mathematical Proficienc = = = 6 Solving Sstems of Equations and Inequalities
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