Solving Systems of Linear Equations

Transcription

1 5 Solving Sstems of Linear Equations 5. Solving Sstems of Linear Equations b Graphing 5. Solving Sstems of Linear Equations b Substitution 5.3 Solving Sstems of Linear Equations b Elimination 5. Solving Special Sstems of Linear Equations 5.5 Solving Equations b Graphing 5.6 Linear Inequalities in Two Variables 5.7 Sstems of Linear Inequalities SEE the Big Idea Fishing (p. 65) Fruit Salad (p. 55) 5) Pets (p. 50) Roofing ofing Contractor t (p. ) ) Drama Club (p. 8) Mathematical Thinking: Mathematicall proficient students can appl the mathematics the know to solve problems arising in everda life, societ, and the workplace.

2 Maintaining Mathematical Proficienc Graphing Linear Functions (A.3.C) Eample Graph 3 + =. Step Rewrite the equation in slope-intercept form. = 3 Step Find the slope and the -intercept. m = and b = 3 Step 3 The -intercept is 3. So, plot (0, 3). Step Use the slope to find another point on the line. slope = rise run = Graph the equation. Plot the point that is units right and unit up from (0, 3). Draw a line through the two points. (0, 3). + =. 6 = = 0. + = 3 Solving and Graphing Linear Inequalities (7.0.B, A.5.B) Eample Solve Graph the solution Write the inequalit Add 5 to each side. 8 Simplif. Subtract from each side. 6 Simplif Divide each side b 6. The solution is. Simplif. Solve the inequalit. Graph the solution m + > t 7. a z + < 9. k 6 < k w + w 3. ABSTRACT REASONING The graphs of the linear functions g and h have different slopes. The value of both functions at = a is b. When g and h are graphed in the same coordinate plane, what happens at the point (a, b)? 7

3 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 Finding the Point of Intersection You can use a graphing calculator to find the point of intersection, if it eists, of the graphs of two linear equations.. Enter the equations into a graphing calculator.. Graph the equations in an appropriate viewing window, so that the point of intersection is visible. 3. Use the intersect feature of the graphing calculator to find the point of intersection. Using a Graphing Calculator Use a graphing calculator to find the point of intersection, if it eists, of the graphs of the two linear equations. = + Equation = 3 5 Equation SOLUTION The slopes of the lines are not the same, so ou know that the lines intersect. Enter the equations into a graphing calculator. Then graph the equations in an appropriate viewing window. 6 = + = Use the intersect feature to find the point of intersection of the lines. 6 6 The point of intersection is (, ). Intersection X= Y= Monitoring Progress Use a graphing calculator to find the point of intersection of the graphs of the two linear equations.. = 3. = = = 3 = = 8 Chapter 5 Solving Sstems of Linear Equations

4 5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..I A.3.F A.3.G A.5.C Solving Sstems of Linear Equations b Graphing Essential Question How can ou solve a sstem of linear equations? Writing a Sstem of Linear Equations Work with a partner. Your famil opens a bed-and-breakfast. The spend $600 preparing a bedroom to rent. The cost to our famil for food and utilities is $5 per night. The charge $75 per night to rent the bedroom. a. Write an equation that represents the costs. Cost, C (in dollars) = $5 per night Number of nights, + $600 b. Write an equation that represents the revenue (income). APPLYING MATHEMATICS To be proficient in math, ou need to identif important quantities in real-life problems and map their relationships using tools such as diagrams, tables, and graphs. Revenue, R (in dollars) = $75 per Number of night nights, c. A set of two (or more) linear equations is called a sstem of linear equations. Write the sstem of linear equations for this problem. Using a Table or Graph to Solve a Sstem Work with a partner. Use the cost and revenue equations from Eploration to determine how man nights our famil needs to rent the bedroom before recovering the cost of preparing the bedroom. This is the break-even point. a. Cop and complete the table. (nights) C (dollars) R (dollars) b. How man nights does our famil need to rent the bedroom before breaking even? c. In the same coordinate plane, graph the cost equation and the revenue equation from Eploration. d. Find the point of intersection of the two graphs. What does this point represent? How does this compare to the break-even point in part (b)? Eplain. Communicate Your Answer 3. How can ou solve a sstem of linear equations? How can ou check our solution?. Solve each sstem b using a table or sketching a graph. Eplain wh ou chose each method. Use a graphing calculator to check each solution. a. =.3.3 b. = c. = = = = Section 5. Solving Sstems of Linear Equations b Graphing 9

5 5. Lesson What You Will Learn Core Vocabular sstem of linear equations, p. 0 solution of a sstem of linear equations, p. 0 Previous linear equation ordered pair Check solutions of sstems of linear equations. Solve sstems of linear equations b graphing. Use sstems of linear equations to solve real-life problems. Sstems of Linear Equations A sstem of linear equations is a set of two or more linear equations in the same variables. An eample is shown below. + = 7 Equation 3 = Equation A solution of a sstem of linear equations in two variables is an ordered pair that is a solution of each equation in the sstem. Checking Solutions Tell whether the ordered pair is a solution of the sstem of linear equations. a. (, 5); + = 7 Equation 3 = Equation b. (, 0); = Equation = + Equation SOLUTION a. Substitute for and 5 for in each equation. READING A sstem of linear equations is also called a linear sstem. Equation Equation + = 7 3 = + 5 =? 7 () 3(5) =? 7 = 7 = Because the ordered pair (, 5) is a solution of each equation, it is a solution of the linear sstem. b. Substitute for and 0 for in each equation. Equation Equation = = + 0 =? ( ) 0 =? + 0 = 0 0 The ordered pair (, 0) is a solution of the first equation, but it is not a solution of the second equation. So, (, 0) is not a solution of the linear sstem. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the ordered pair is a solution of the sstem of linear equations.. (, ); + = 0 + = 5. (, ); = 3 + = Chapter 5 Solving Sstems of Linear Equations

6 REMEMBER Note that the linear equations are in slope-intercept form. You can use the method presented in Section 3.5 to graph the equations. Solving Sstems of Linear Equations b Graphing The solution of a sstem of linear equations is the point of intersection of the graphs of the equations. Core Concept Solving a Sstem of Linear Equations b Graphing Step Graph each equation in the same coordinate plane. Step Estimate the point of intersection. Step 3 Check the point from Step b substituting for and in each equation of the original sstem. Solving a Sstem of Linear Equations b Graphing Solve the sstem of linear equations b graphing. = + 5 Equation = Equation SOLUTION Step Graph each equation. Step Estimate the point of intersection. The graphs appear to intersect at (, 3). Step 3 Check our point from Step. Equation Equation = + 5 = (, 3) = + 5 = 3 =? () =? () 3 = 3 3 = 3 The solution is (, 3). Check 6 = + 5 = 6 Intersection X= Y=3 6 Monitoring Progress Solve the sstem of linear equations b graphing. Help in English and Spanish at BigIdeasMath.com 3. =. = = 5 = + = = Section 5. Solving Sstems of Linear Equations b Graphing

7 Solving Real-Life Problems Modeling with Mathematics A roofing contractor bus 30 bundles of shingles and rolls of roofing paper for $00. In a second purchase (at the same prices), the contractor bus 8 bundles of shingles for $56. Find the price per bundle of shingles and the price per roll of roofing paper. SOLUTION. Understand the Problem You know the total price of each purchase and how man of each item were purchased. You are asked to find the price of each item.. Make a Plan Use a verbal model to write a sstem of linear equations that represents the problem. Then solve the sstem of linear equations. 3. Solve the Problem Words 30 8 Price per bundle Price per bundle + Price per roll + 0 Price per roll = 00 = 56 Variables Let be the price (in dollars) per bundle and let be the price (in dollars) per roll. Sstem 30 + = 00 Equation 8 = 56 Equation Step Graph each equation. Note that onl the first quadrant is shown because and must be positive. Step Estimate the point of intersection. The graphs appear to intersect at (3, 0). Step 3 Check our point from Step. Equation Equation 30 + = 00 8 = 56 30(3) + (0) =? 00 8(3) =? = = 56 The solution is (3, 0). So, the price per bundle of shingles is $3, and the price per roll of roofing paper is $0.. Look Back You can use estimation to check that our solution is reasonable. A bundle of shingles costs about $30. So, 30 bundles of shingles and rolls of roofing paper (at $0 per roll) cost about 30(30) + (0) = $980, and 8 bundles of shingles costs about 8(30) = $0. These prices are close to the given values, so the solution seems reasonable = = 3 (3, 0) Monitoring Progress Help in English and Spanish at BigIdeasMath.com 6. You have a total of 8 math and science eercises for homework. You have si more math eercises than science eercises. How man eercises do ou have in each subject? Chapter 5 Solving Sstems of Linear Equations

8 5. Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. VOCABULARY Do the equations 5 = 8 and 6 = 0 form a sstem of linear equations? Eplain.. DIFFERENT WORDS, SAME QUESTION Consider the sstem of linear equations + = and = 6. Which is different? Find both answers. Solve the sstem of linear equations. Solve each equation for. Find the point of intersection of the graphs of the equations. Find an ordered pair that is a solution of each equation in the sstem. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, tell whether the ordered pair is a solution of the sstem of linear equations. (See Eample.) 3. (, 6); + = 8 3 = 0 5. (, 3); = 7 = (, ); = + 6 = 3 7. (, ); = 7 = 8. (8, ); = 6 0 = 8. (5, 6); = + = In Eercises 9, use the graph to solve the sstem of linear equations. Check our solution. 9. = 0. + = 5 + = = In Eercises 3 0, solve the sstem of linear equations b graphing. (See Eample.) 3. = + 7. = + = + = 8 5. = = 3 = = = 3 8. = 0 = = 5 9. = = 3 3 = + 3 = 6 ERROR ANALYSIS In Eercises and, describe and correct the error in solving the sstem of linear equations.. The solution of the linear sstem 3 = 6 and 3 = 3 is (3, ) = 8. = + = + = 8 6. The solution of the linear sstem = and = + is =. Section 5. Solving Sstems of Linear Equations b Graphing 3

9 USING TOOLS In Eercises 3 6, use a graphing calculator to solve the sstem of linear equations = = = = = 0 6. = = + =.5 7. MODELING WITH MATHEMATICS You have 0 minutes to eercise at the gm, and ou want to burn 300 calories total using both machines. How much time should ou spend on each machine? (See Eample 3.) Elliptical Trainer 8 calories per minute 8. MODELING WITH MATHEMATICS You sell small and large candles at a craft fair. You collect $ selling a total of 8 candles. How man of each tpe of candle did ou sell? Stationar Bike 6 calories per minute $6 each $ each 9. MATHEMATICAL CONNECTIONS Write a linear equation that represents the area and a linear equation that represents the perimeter of the rectangle. Solve the sstem of linear equations b graphing. Interpret our solution. 3. COMPARING METHODS Consider the equation + = 3. a. Solve the equation using algebra. b. Solve the sstem of linear equations = + and = 3 b graphing. c. How is the linear sstem and the solution in part (b) related to the original equation and the solution in part (a)? 3. HOW DO YOU SEE IT? A teacher is purchasing binders for students. The graph shows the total costs of ordering binders from three different companies. Cost (dollars) Buing Binders Compan A Compan B Compan C Number of binders a. For what numbers of binders are the costs the same at two different companies? Eplain. b. How do our answers in part (a) relate to sstems of linear equations? 33. MAKING AN ARGUMENT You and a friend are going hiking but start at different locations. You start at the trailhead and walk 5 miles per hour. Your friend starts 3 miles from the trailhead and walks 3 miles per hour. ou (3 3) cm 6 cm 30. THOUGHT PROVOKING Your friend s bank account balance (in dollars) is represented b the equation = , where is the number of months. Graph this equation. After 6 months, ou want to have the same account balance as our friend. Write a linear equation that represents our account balance. Interpret the slope and -intercept of the line that represents our account balance. Maintaining Mathematical Proficienc Solve the literal equation for. (Section.) = = Chapter 5 Solving Sstems of Linear Equations Reviewing what ou learned in previous grades and lessons 3 + = 5 our friend a. Write and graph a sstem of linear equations that represents this situation. b. Your friend sas that after an hour of hiking ou will both be at the same location on the trail. Is our friend correct? Use the graph from part (a) to eplain our answer.

10 5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..I A.5.C Solving Sstems of Linear Equations b Substitution Essential Question How can ou use substitution to solve a sstem of linear equations? Using Substitution to Solve Sstems Work with a partner. Solve each sstem of linear equations using two methods. Method Solve for first. Solve for in one of the equations. Substitute the epression for into the other equation to find. Then substitute the value of into one of the original equations to find. Method Solve for first. Solve for in one of the equations. Substitute the epression for into the other equation to find. Then substitute the value of into one of the original equations to find. Is the solution the same using both methods? Eplain which method ou would prefer to use for each sstem. a. + = 7 b. 6 = c. + = 5 + = = = 8 Writing and Solving a Sstem of Equations USING PRECISE MATHEMATICAL LANGUAGE To be proficient in math, ou need to communicate precisel with others. Work with a partner. a. Write a random ordered pair with integer coordinates. One wa to do this is to use a graphing calculator. The ordered pair generated at the right is (, 3). b. Write a sstem of linear equations that has our ordered pair as its solution. c. Echange sstems with our partner and use one of the methods from Eploration to solve the sstem. Eplain our choice of method. Communicate Your Answer Choose two random integers between 5 and 5. randint(-5 5 ) {- -3} 3. How can ou use substitution to solve a sstem of linear equations?. Use one of the methods from Eploration to solve each sstem of linear equations. Eplain our choice of method. Check our solutions. a. + = 7 b. = 6 c. 3 + = 0 = 9 + = + = 6 d. 3 + = 3 e. 3 = 9 f. 3 = 6 3 = 3 3 = = Section 5. Solving Sstems of Linear Equations b Substitution 5

11 5. Lesson What You Will Learn Core Vocabular Previous sstem of linear equations solution of a sstem of linear equations Solve sstems of linear equations b substitution. Use sstems of linear equations to solve real-life problems. Solving Linear Sstems b Substitution Another wa to solve a sstem of linear equations is to use substitution. Core Concept Solving a Sstem of Linear Equations b Substitution Step Solve one of the equations for one of the variables. Step Substitute the epression from Step into the other equation and solve for the other variable. Step 3 Substitute the value from Step into one of the original equations and solve. Solving a Sstem of Linear Equations b Substitution Solve the sstem of linear equations b substitution. = 9 Equation 6 5 = 9 Equation SOLUTION Step Equation is alread solved for. Step Substitute 9 for in Equation and solve for. Check Equation = 9 =? ( ) 9 = Equation 6 5 = 9 6( ) 5( ) =? 9 9 = = 9 Equation 6 5( 9) = 9 Substitute 9 for = 9 Distributive Propert = 9 Combine like terms. 6 = 6 Subtract 5 from each side. = Divide each side b 6. Step 3 Substitute for in Equation and solve for. = 9 Equation = ( ) 9 Substitute for. = 8 9 Multipl. = Subtract. The solution is (, ). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem of linear equations b substitution. Check our solution.. = = 0 3. = 6 7 = = + = 3 6 Chapter 5 Solving Sstems of Linear Equations

12 ANOTHER WAY You could also begin b solving for in Equation, solving for in Equation, or solving for in Equation. Solving a Sstem of Linear Equations b Substitution Solve the sstem of linear equations b substitution. + = 3 Equation 3 + = Equation SOLUTION Step Solve for in Equation. = + 3 Revised Equation Step Substitute + 3 for in Equation and solve for. 3 + = Equation 3 + ( + 3) = Substitute + 3 for. + 3 = = Combine like terms. Subtract 3 from each side. = Divide each side b. Step 3 Substitute for in Equation and solve for. + = 3 Equation ( ) + = 3 Substitute for. = Subtract from each side. The solution is (, ). Algebraic Check Equation + = 3 ( ) + =? 3 3 = 3 Equation 3 + = Graphical Check = + 3 = 3 5 Intersection X=- Y= 3( ) + =? = Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem of linear equations b substitution. Check our solution.. + = 5. + = 3 + = 6 = 0 6. = 5 7. = 7 3 = 3 = 3 Section 5. Solving Sstems of Linear Equations b Substitution 7

13 Solving Real-Life Problems Modeling with Mathematics A drama club earns $00 from a production. An adult ticket costs twice as much as a student ticket. Write a sstem of linear equations that represents this situation. What is the price of each tpe of ticket? SOLUTION. Understand the Problem You know the amount earned, the total numbers of adult and student tickets sold, and the relationship between the price of an adult ticket and the price of a student ticket. You are asked to write a sstem of linear equations that represents the situation and find the price of each tpe of ticket.. Make a Plan Use a verbal model to write a sstem of linear equations that represents the problem. Then solve the sstem of linear equations. 3. Solve the Problem Words 6 Adult ticket price Adult ticket price + 3 Student ticket price = 00 = Student ticket price Variables Let be the price (in dollars) of an adult ticket and let be the price (in dollars) of a student ticket. Sstem = 00 Equation = Equation Step Equation is alread solved for. Step Substitute for in Equation and solve for. Tickets sold Tpe Number adult 6 student 3 STUDY TIP You can use either of the original equations to solve for. However, using Equation requires fewer calculations = 00 Equation 6() + 3 = 00 Substitute for. 60 = 00 Simplif. = Simplif. Step 3 Substitute for in Equation and solve for. = Equation = () Substitute for. = 8 Simplif. The solution is (8, ). So, an adult ticket costs $8 and a student ticket costs $.. Look Back To check that our solution is correct, substitute the values of and into both of the original equations and simplif. Monitoring Progress 6(8) + 3() = 00 8 = () 00 = 00 8 = 8 Help in English and Spanish at BigIdeasMath.com 8. There are a total of 6 students in a drama club and a earbook club. The drama club has 0 more students than the earbook club. Write a sstem of linear equations that represents this situation. How man students are in each club? 8 Chapter 5 Solving Sstems of Linear Equations

14 5. Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. WRITING Describe how to solve a sstem of linear equations b substitution.. NUMBER SENSE When solving a sstem of linear equations b substitution, how do ou decide which variable to solve for in Step? Monitoring Progress and Modeling with Mathematics In Eercises 3 8, tell which equation ou would choose to solve for one of the variables. Eplain = = 0 = 0 + = = 6. 3 = 9 5 = 5 + = 8 7. = = = 5 = 6 In Eercises 9 6, solve the stem of linear equations b substitution. Check our solution. (See Eamples and.) 9. = = = = 3. = = = 8 = =. = 3 5 = 9 9 = = = + = 3 = 7. ERROR ANALYSIS Describe and correct the error in solving for one of the variables in the linear sstem 8 + = and 5 =. 8. ERROR ANALYSIS Describe and correct the error in solving for one of the variables in the linear sstem + = 6 and 3 + = 9. Step 3 + = 9 = 9 3 Step + (9 3) = = 6 = = 6 Step = = 9 3 = 3 = 9. MODELING WITH MATHEMATICS A farmer plants corn and wheat on a 80-acre farm. The farmer wants to plant three times as man acres of corn as wheat. Write a sstem of linear equations that represents this situation. How man acres of each crop should the farmer plant? (See Eample 3.) 0. MODELING WITH MATHEMATICS A compan that offers tubing trips down a river rents tubes for a person to use and cooler tubes to carr food and water. A group spends $70 to rent a total of 5 tubes. Write a sstem of linear equations that represents this situation. How man of each tpe of tube does the group rent? Step 5 = = 5 + = 5 Step 5 (5 ) = = = Section 5. Solving Sstems of Linear Equations b Substitution 9

15 In Eercises, write a sstem of linear equations that has the ordered pair as its solution.. (3, 5). (, 8) 3. (, ). (5, 5) 5. PROBLEM SOLVING A math test is worth 00 points and has 38 problems. Each problem is worth either 5 points or points. How man problems of each point value are on the test? 6. PROBLEM SOLVING An investor owns shares of Stock A and Stock B. The investor owns a total of 00 shares with a total value of $000. How man shares of each stock does the investor own? Stock Price A $9.50 B $ MAKING AN ARGUMENT Your friend sas that given a linear sstem with an equation of a horizontal line and an equation of a vertical line, ou cannot solve the sstem b substitution. Is our friend correct? Eplain. 3. OPEN-ENDED Write a sstem of linear equations in which (3, 5) is a solution of Equation but not a solution of Equation, and (, 7) is a solution of the sstem. 3. HOW DO YOU SEE IT? The graphs of two linear equations are shown. 6 = + = 6 MATHEMATICAL CONNECTIONS In Eercises 7 and 8, (a) write an equation that represents the sum of the angle measures of the triangle and (b) use our equation and the equation shown to find the values of and a. At what point do the lines appear to intersect? b. Could ou solve a sstem of linear equations b substitution to check our answer in part (a)? Eplain = 3 ( 8) 3 5 = 33. REPEATED REASONING A radio station plas a total of 7 pop, rock, and hip-hop songs during a da. The number of pop songs is 3 times the number of rock songs. The number of hip-hop songs is 3 more than the number of rock songs. How man of each tpe of song does the radio station pla? 9. REASONING Find the values of a and b so that the solution of the linear sstem is ( 9, ). a + b = 3 Equation a b = Equation Maintaining Mathematical Proficienc 3. THOUGHT PROVOKING You have $.65 in coins. Write a sstem of equations that represents this situation. Use variables to represent the number of each tpe of coin. 35. NUMBER SENSE The sum of the digits of a two-digit number is. When the digits are reversed, the number increases b 7. Find the original number. Reviewing what ou learned in previous grades and lessons Find the sum or difference. (Skills Review Handbook) 36. ( ) + ( 7) 37. (5 ) + ( 5 ) 38. (t 8) (t + 5) 39. (6d + ) (3d 3) 0. (m + ) + 3(6m ). (5v + 6) 6( 9v + ) 30 Chapter 5 Solving Sstems of Linear Equations

16 5.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..I A.5.C Solving Sstems of Linear Equations b Elimination Essential Question How can ou use elimination to solve a sstem of linear equations? Writing and Solving a Sstem of Equations Work with a partner. You purchase a drink and a sandwich for $.50. Your friend purchases a drink and five sandwiches for $6.50. You want to determine the price of a drink and the price of a sandwich. a. Let represent the price (in dollars) of one drink. Let represent the price (in dollars) of one sandwich. Write a sstem of equations for the situation. Use the following verbal model. Number Price of drinks per drink Number of Price per + sandwiches sandwich = Total price Label one of the equations Equation and the other equation Equation. USING PROBLEM-SOLVING STRATEGIES To be proficient in math, ou need to monitor and evaluate our progress and change course using a different solution method, if necessar. b. Subtract Equation from Equation. Eplain how ou can use the result to solve the sstem of equations. Then find and interpret the solution. Using Elimination to Solve Sstems Work with a partner. Solve each sstem of linear equations using two methods. Method Subtract. Subtract Equation from Equation. Then use the result to solve the sstem. Method Add. Add the two equations. Then use the result to solve the sstem. Is the solution the same using both methods? Which method do ou prefer? a. 3 = 6 b. + = 6 c. = = 0 = + = 5 Work with a partner. + = 7 Equation + 5 = 7 Equation Using Elimination to Solve a Sstem a. Can ou eliminate a variable b adding or subtracting the equations as the are? If not, what do ou need to do to one or both equations so that ou can? b. Solve the sstem individuall. Then echange solutions with our partner and compare and check the solutions. Communicate Your Answer. How can ou use elimination to solve a sstem of linear equations? 5. When can ou add or subtract the equations in a sstem to solve the sstem? When do ou have to multipl first? Justif our answers with eamples. 6. In Eploration 3, wh can ou multipl an equation in the sstem b a constant and not change the solution of the sstem? Eplain our reasoning. Section 5.3 Solving Sstems of Linear Equations b Elimination 3

17 5.3 Lesson What You Will Learn Core Vocabular Previous coefficient Solve sstems of linear equations b elimination. Use sstems of linear equations to solve real-life problems. Solving Linear Sstems b Elimination Core Concept Solving a Sstem of Linear Equations b Elimination Step Multipl, if necessar, one or both equations b a constant so at least one pair of like terms has the same or opposite coefficients. Step Add or subtract the equations to eliminate one of the variables. Step 3 Solve the resulting equation. Step Substitute the value from Step 3 into one of the original equations and solve for the other variable. You can use elimination to solve a sstem of equations because replacing one equation in the sstem with the sum of that equation and a multiple of the other produces a sstem that has the same solution. Here is wh. Sstem a = b Equation c = d Equation Sstem a + kc = b + kd Equation 3 c = d Equation Consider Sstem. In this sstem, a and c are algebraic epressions, and b and d are constants. Begin b multipling each side of Equation b a constant k. B the Multiplication Propert of Equalit, kc = kd. You can rewrite Equation as Equation 3 b adding kc on the left and kd on the right. You can rewrite Equation 3 as Equation b subtracting kc on the left and kd on the right. Because ou can rewrite either sstem as the other, Sstem and Sstem have the same solution. Solving a Sstem of Linear Equations b Elimination Solve the sstem of linear equations b elimination. 3 + = Equation 3 = Equation SOLUTION Step Because the coefficients of the -terms are opposites, ou do not need to multipl either equation b a constant. Step Add the equations. Check Equation 3 + = 3(0) + () =? = Equation 3 = 3(0) () =? = 3 + = Equation 3 = Equation 6 = 0 Add the equations. Step 3 Solve for. 6 = 0 Resulting equation from Step = 0 Divide each side b 6. Step Substitute 0 for in one of the original equations and solve for. 3 + = Equation 3(0) + = Substitute 0 for. = Solve for. The solution is (0, ). 3 Chapter 5 Solving Sstems of Linear Equations

18 ANOTHER WAY To use subtraction to eliminate one of the variables, multipl Equation b and then subtract the equations = ( 0 = 6) 5 = 5 Check 0 0 Intersection X=- Y=-3 0 Equation 0 Equation Solving a Sstem of Linear Equations b Elimination Solve the sstem of linear equations b elimination = Equation 5 6 = 3 Equation SOLUTION Step Multipl Equation b so that the coefficients of the -terms are opposites = = Equation 5 6 = 3 Multipl b. 0 + = 6 Revised Equation Step Add the equations = Equation 0 + = 6 Revised Equation 5 = 5 Add the equations. Step 3 Solve for. 5 = 5 Resulting equation from Step = 3 Divide each side b 5. Step Substitute 3 for in one of the original equations and solve for. 5 6 = 3 Equation 5 6( 3) = 3 Substitute 3 for = 3 5 = 5 The solution is (, 3). Multipl. Subtract 8 from each side. = Divide each side b 5. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem of linear equations b elimination. Check our solution = 7. 3 = 3. + = 3 + = = + = 3 Concept Summar Methods for Solving Sstems of Linear Equations Method Graphing (Lesson 5.) Substitution (Lesson 5.) Elimination (Lesson 5.3) Elimination (Multipl First) (Lesson 5.3) To estimate solutions When to Use When one of the variables in one of the equations has a coefficient of or When at least one pair of like terms has the same or opposite coefficients When one of the variables cannot be eliminated b adding or subtracting the equations Section 5.3 Solving Sstems of Linear Equations b Elimination 33

19 Average salar (thousands of dollars) Classroom Teacher (0, 5) District A District B (5, 5) (5, 60) (5, 55) Years since 985 Solving Real-Life Problems Modeling with Mathematics The graph represents the average salaries of classroom teachers in two school districts. During what ear were the average salaries in the two districts equal? What was the average salar in both districts in that ear? SOLUTION. Understand the Problem You know two points on each line in the graph. You are asked to determine the ear in which the average salaries were equal and then determine the average salar in that ear.. Make a Plan Use the points in the graph to write a sstem of linear equations. Then solve the sstem of linear equations. 3. Solve the Problem Find the slope of each line District A: m = 5 5 = 35 0 = District B: m = 5 0 = 30 5 = 6 5 Use each slope and a point on each line to write equations of the lines. District A District B = m( ) Write the point-slope form. = m( ) STUDY TIP In Eample 3, both equations are multiplied b a constant so that the coefficients of the -terms are opposites. 6 8 (, 9) Line B (8, 6) Line A (, 5) 5 = 7 ( 5) Substitute for m,, and. 5 = 6 ( 0) = 65 Write in standard form = 5 Sstem 7 + = 65 Equation = 5 Equation Step Multipl Equation b 5. Multipl Equation b. 7 + = 65 Multipl b = 35 Revised Equation = 5 Multipl b. + 0 = 500 Revised Equation Step Add the equations = 35 Revised Equation + 0 = 500 Revised Equation = 75 Add the equations. Step 3 Solving the equation = 75 gives = Step Substitute 75 for in one of the original equations and solve for. 7 + = 65 Equation 7 ( 75 ) + = 65 Substitute 75 Solve for. for. The solution is about (5.9, ). Because 5.9 corresponds to the ear 000, the average salar in both districts was about $,000 in Look Back Using the graph, the point of intersection appears to be about (5, 5). So, the solution of (5.9, ) is reasonable. 0 0 (, ) 8 6 Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Write and solve a sstem of linear equations represented b the graph at the left. 3 Chapter 5 Solving Sstems of Linear Equations

20 5.3 Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. OPEN-ENDED Give an eample of a sstem of linear equations that can be solved b first adding the equations to eliminate one variable.. WRITING Eplain how to solve the sstem of linear equations 3 = Equation b elimination = 7 Equation Monitoring Progress and Modeling with Mathematics In Eercises 3 0, solve the sstem of linear equations b elimination. Check our solution. (See Eample.) 3. + = = + = 5 = = = 6 = = = = + 5 = 3 = = = 5 + = = 3 In Eercises 8, solve the sstem of linear equations b elimination. Check our solution. (See Eamples and 3.). + =. 8 5 = + 7 = 9 3 = = = = = = = 9 5 = 3 + = = = = = ERROR ANALYSIS Describe and correct the error in solving for one of the variables in the linear sstem 5 7 = 6 and + 7 = = = 8 = = 6 0. ERROR ANALYSIS Describe and correct the error in solving for one of the variables in the linear sstem + 3 = 8 and = = = 8 = 3 Multipl b. + 8 = 3. MODELING WITH MATHEMATICS A service center charges a fee of dollars for an oil change plus dollars per quart of oil used. A sample of its sales record is shown. Write a sstem of linear equations that represents this situation. Find the fee and cost per quart of oil. 3 A B Oil Tank Size Customer (quarts) A 5 B 7. MODELING WITH MATHEMATICS The graph represents the average salaries of high school principals in two states. During what ear were the average salaries in the two states equal? What was the average salar in both states in that ear? Average salar (thousands of dollars) (0, 3) = 5 = 5 C Total Cost $.5 $5.5 High School Principal State A State B (, 00) (6, 8) (5, 98) Years since 985 Section 5.3 Solving Sstems of Linear Equations b Elimination 35

21 In Eercises 3 6, solve the sstem of linear equations using an method. Eplain wh ou chose the method =. 6 + = = 8 5 = 5. = = 3 = = WRITING For what values of a can ou solve the linear sstem a + 3 = and + 5 = 6 b elimination without multipling first? Eplain. 30. THOUGHT PROVOKING Write a sstem of linear equations that can be added to eliminate a variable or subtracted to eliminate a variable. 3. MATHEMATICAL CONNECTIONS A rectangle has a perimeter of 8 inches. A new rectangle is formed b doubling the width w and tripling the length, as shown. The new rectangle has a perimeter P of 6 inches. P = 6 in. w 8. HOW DO YOU SEE IT? The circle graph shows the results of a surve in which 50 students were asked about their favorite meal. Favorite Meal Dinner 5 Breakfast Lunch a. Estimate the numbers of students who chose breakfast and lunch. b. The number of students who chose lunch was 5 more than the number of students who chose breakfast. Write a sstem of linear equations that represents the numbers of students who chose breakfast and lunch. c. Eplain how ou can solve the linear sstem in part (b) to check our answers in part (a). 9. MAKING AN ARGUMENT Your friend sas that an sstem of equations that can be solved b elimination can be solved b substitution in an equal or fewer number of steps. Is our friend correct? Eplain. Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons 3 a. Write and solve a sstem of linear equations to find the length and width of the original rectangle. b. Find the length and width of the new rectangle. 3. CRITICAL THINKING Refer to the discussion of Sstem and Sstem on page 3. Without solving, eplain wh the two sstems shown have the same solution. Sstem Sstem 3 = 8 Equation 5 = 0 Equation 3 + = 6 Equation + = 6 Equation 33. PROBLEM SOLVING You are making 6 quarts of fruit punch for a part. You have bottles of 00% fruit juice and 0% fruit juice. How man quarts of each tpe of juice should ou mi to make 6 quarts of 80% fruit juice? 3. PROBLEM SOLVING A motorboat takes 0 minutes to travel 0 miles downstream. The return trip takes 60 minutes. What is the speed of the current? 35. CRITICAL THINKING Solve for,, and z in the sstem of equations. Eplain our steps z = 9 Equation 3z + = 7 Equation 5 = 0 Equation 3 Solve the equation. Determine whether the equation has one solution, no solution, or infinitel man solutions. (Section.3) 36. 5d 8 = + 5d t = t 38. 3n + = (n 3) 39. 3( v) = 6v Write an equation of the line that passes through the given point and is parallel to the given line. (Section.) 0. (, ); = + 7. (0, 6); = 5 3. ( 5, ); = Chapter 5 Solving Sstems of Linear Equations

22 5. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..I A.3.F A.5.C Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest $50 for equipment to make skateboards. The materials for each skateboard cost $0. You sell each skateboard for $0. a. Write the cost and revenue equations. Then cop and complete the table for our cost C and our revenue R. (skateboards) C (dollars) R (dollars) APPLYING MATHEMATICS To be proficient in math, ou need to interpret mathematical results in real-life contets. b. When will our compan break even? What is wrong? Writing and Analzing a Sstem Work with a partner. A necklace and matching bracelet have two tpes of beads. The necklace has 0 small beads and 6 large beads and weighs 0 grams. The bracelet has 0 small beads and 3 large beads and weighs 5 grams. The threads holding the beads have no significant weight. a. Write a sstem of linear equations that represents the situation. Let be the weight (in grams) of a small bead and let be the weight (in grams) of a large bead. b. Graph the sstem in the coordinate plane shown. What do ou notice about the two lines? c. Can ou find the weight of each tpe of bead? Eplain our reasoning Communicate Your Answer Can a sstem of linear equations have no solution or infinitel man solutions? Give eamples to support our answers.. Does the sstem of linear equations represented b each graph have no solution, one solution, or infi nitel man solutions? Eplain. a. = + b. 6 = + c. 6 = + + = 3 + = 3 + = Section 5. Solving Special Sstems of Linear Equations 37

23 5. Lesson What You Will Learn Core Vocabular Previous parallel Determine the numbers of solutions of linear sstems. Use linear sstems to solve real-life problems. The Numbers of Solutions of Linear Sstems Core Concept Solutions of Sstems of Linear Equations A sstem of linear equations can have one solution, no solution, or infi nitel man solutions. One solution No solution Infinitel man solutions ANOTHER WAY You can solve some linear sstems b inspection. In Eample, notice ou can rewrite the sstem as + = + = 5. This sstem has no solution because + cannot be equal to both and 5. The lines intersect. The lines are parallel. The lines are the same. Solving a Sstem: No Solution Solve the sstem of linear equations. = + Equation = 5 Equation SOLUTION Method Solve b graphing. Graph each equation. The lines have the same slope and different -intercepts. So, the lines are parallel. = + Because parallel lines do not intersect, there is no point that is a solution of both equations. = 5 So, the sstem of linear equations has no solution. Method Solve b substitution. Substitute 5 for in Equation. STUDY TIP A linear sstem with no solution is called an inconsistent sstem. = + Equation 5 = + Substitute 5 for. 5 = Subtract from each side. The equation 5 = is never true. So, the sstem of linear equations has no solution. 38 Chapter 5 Solving Sstems of Linear Equations

24 ANOTHER WAY You can also solve the linear sstem b graphing. The lines have the same slope and the same -intercept. So, the lines are the same, which means all points on the line are solutions of both equations. Solving a Sstem: Infinitel Man Solutions Solve the sstem of linear equations. + = 3 Equation + = 6 Equation SOLUTION Solve b elimination. Step Multipl Equation b. + = 3 Multipl b. = 6 Revised Equation + = 6 + = 6 Equation STUDY TIP A linear sstem with infinitel man solutions is called a consistent dependent sstem. Step Add the equations. = 6 Revised Equation + = 6 Equation 0 = 0 Add the equations. The equation 0 = 0 is alwas true. So, the solutions are all the points on the line + = 3. The sstem of linear equations has infinitel man solutions. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem of linear equations.. + = 3. = = 6 + = Solving Real-Life Problems Modeling with Mathematics Cost (dollars) Hours Hall A Hall B An athletic director is comparing the costs of renting two banquet halls for an awards banquet. Write a sstem of linear equations that represents this situation. If the cost patterns continue, will the cost of Hall A ever equal the cost of Hall B? SOLUTION Words Total cost = Cost per hour Number of hours + Initial cost Variables Let be the cost (in dollars) and let be the number of hours. Sstem = Equation - Cost of Hall A = Equation - Cost of Hall B The equations are in slope-intercept form. The graphs of the equations have the same slope but different -intercepts. There is no solution because the lines are parallel. So, the cost of Hall A will never equal the cost of Hall B. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 3. WHAT IF? What happens to the solution in Eample 3 when the cost per hour for Hall A is $5? Section 5. Solving Special Sstems of Linear Equations 39

25 Modeling with Mathematics 6 6 The perimeter of the trapezoidal piece of land is 8 kilometers. The perimeter of the rectangular piece of land is kilometers. Write and solve a sstem of linear equations to find the values of and. 8 SOLUTION. Understand the Problem You know the perimeter of each piece of land and the side lengths in terms of or. You are asked to write and solve a sstem of linear equations to find the values of and Make a Plan Use the figures and the definition of perimeter to write a sstem of linear equations that represents the problem. Then solve the sstem of linear equations Solve the Problem Perimeter of trapezoid Perimeter of rectangle = = 6 + = 8 Equation = Equation Sstem 6 + = 8 Equation = Equation Method Solve b graphing. Graph each equation. The lines have the same slope and the same -intercept. So, the lines are the same. In this contet, and must be positive. Because the lines are the same, all the points on the line in Quadrant I are solutions of both equations = 6 + = So, the sstem of linear equations has infinitel man solutions. Method Solve b elimination. Multipl Equation b 3 and add the equations. 6 + = 8 Multipl b = Revised Equation = = Equation 0 = 0 Add the equations. The equation 0 = 0 is alwas true. In this contet, and must be positive. So, the solutions are all the points on the line 6 + = 8 in Quadrant I. The sstem of linear equations has infinitel man solutions.. Look Back Choose a few of the ordered pairs (, ) that are solutions of Equation. You should find that no matter which ordered pairs ou choose, the will also be solutions of Equation. So, infi nitel man solutions seems reasonable. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? What happens to the solution in Eample when the perimeter of the trapezoidal piece of land is 96 kilometers? Eplain. 0 Chapter 5 Solving Sstems of Linear Equations

26 5. Eercises Tutorial Help in English and Spanish at BigIdeasMath.com Vocabular and Core Concept Check. REASONING Is it possible for a sstem of linear equations to have eactl two solutions? Eplain.. WRITING Compare the graph of a sstem of linear equations that has infinitel man solutions and the graph of a sstem of linear equations that has no solution. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, match the sstem of linear equations with its graph. Then determine whether the sstem has one solution, no solution, or infinitel man solutions. A =. = = + = 5. + = 6. = 0 = 8 5 = = = = 9 3 = B = = 0 = 3 + = = 6. 3 = = = 7 In Eercises 7, use onl the slopes and -intercepts of the graphs of the equations to determine whether the sstem of linear equations has one solution, no solution, or infinitel man solutions. Eplain. 7. = = = 39 + = = = 3 = 7 = 8 C. D =. = 6 3 = 3 6 = 30 ERROR ANALYSIS In Eercises 3 and, describe and correct the error in solving the sstem of linear equations. E. F = + = 3 3 In Eercises 9 6, solve the sstem of linear equations. (See Eamples and.) 3 9. = 0. = 6 8 = = = 6. + = = 6 = 7. The lines do not intersect. So, the sstem has no solution. = 3 8 = 3 The lines have the same slope. So, the sstem has infinitel man solutions. Section 5. Solving Special Sstems of Linear Equations

27 5. MODELING WITH MATHEMATICS The table shows the distances two groups have traveled at different times during a canoeing ecursion. The groups continue traveling at their current rates until the reach the same destination. Let d be the distance traveled and t be the time since p.m. Write a sstem of linear equations that represents this situation. Will Group B catch up to Group A before reaching the destination? Eplain. (See Eample 3.) Distance Traveled (miles) P.M. P.M. 3 P.M. P.M. Group A Group B MODELING WITH MATHEMATICS A $6-bag of trail mi contains 3 cups of dried fruit and cups of almonds. A $9-bag contains cups of dried fruit and 6 cups of almonds. Write and solve a sstem of linear equations to find the price of cup of dried fruit and cup of almonds. (See Eample.) 7. PROBLEM SOLVING A train travels from New York Cit to Washington, D.C., and then back to New York Cit. The table shows the number of tickets purchased for each leg of the trip. The cost per ticket is the same for each leg of the trip. Is there enough information to determine the cost of one coach ticket? Eplain. Destination Coach tickets Business class tickets Mone collected (dollars) Washington, D.C ,860 New York Cit ,80 8. THOUGHT PROVOKING Write a sstem of three linear equations in two variables so that an two of the equations have eactl one solution, but the entire sstem of equations has no solution. 9. REASONING In a sstem of linear equations, one equation has a slope of and the other equation has a slope of 3. How man solutions does the sstem have? Eplain. 30. HOW DO YOU SEE IT? The graph shows information about the last leg of a 00-meter rela for three rela teams. Team A s runner ran about 7.8 meters per second, Team B s runner ran about 7.8 meters per second, and Team C s runner ran about 8.8 meters per second. Distance (meters) Last Leg of 00-Meter Rela Team A Team C Team B Time (seconds) a. Estimate the distance at which Team C s runner passed Team B s runner. b. If the race was longer, could Team C s runner have passed Team A s runner? Eplain. c. If the race was longer, could Team B s runner have passed Team A s runner? Eplain. 3. ABSTRACT REASONING Consider the sstem of linear equations = a + and = b, where a and b are real numbers. Determine whether each statement is alwas, sometimes, or never true. Eplain our reasoning. a. The sstem has infinitel man solutions. b. The sstem has no solution. c. When a < b, the sstem has one solution. 3. MAKING AN ARGUMENT One admission to an ice skating rink costs dollars, and renting a pair of ice skates costs dollars. Your friend sas she can determine the eact cost of one admission and one skate rental. Is our friend correct? Eplain. 3 Admissions Skate Rentals 5 Admissions 0 Skate Rentals Total $ Total $ Maintaining Mathematical Proficienc Solve the sstem of linear equations b graphing. (Section 5.) Reviewing what ou learned in previous grades and lessons 33. = 6 3. = = 6 = + 0 = = 6 Chapter 5 Solving Sstems of Linear Equations

28 5. 5. What Did You Learn? Core Vocabular sstem of linear equations, p. 0 solution of a sstem of linear equations, p. 0 Core Concepts Section 5. Solving a Sstem of Linear Equations b Graphing, p. Section 5. Solving a Sstem of Linear Equations b Substitution, p. 6 Section 5.3 Solving a Sstem of Linear Equations b Elimination, p. 3 Section 5. Solutions of Sstems of Linear Equations, p. 38 Mathematical Thinking. Describe the given information in Eercise 33 on page 30 and our plan for finding the solution.. Describe another real-life situation similar to Eercise on page 35 and the mathematics that ou can appl to solve the problem. 3. What question(s) can ou ask our friend to help her understand the error in the statement she made in Eercise 3 on page? Stud Skills Analzing Your Errors Stud Errors What Happens: You do not stud the right material or ou do not learn it well enough to remember it on a test without resources such as notes. How to Avoid This Error: Take a practice test. Work with a stud group. Discuss the topics on the test with our teacher. Do not tr to learn a whole chapter s worth of material in one night. 3

29 5. 5. Quiz Use the graph to solve the sstem of linear equations. Check our solution. (Section 5.). = 3 +. = 3. = = = + 6 = + 3 Solve the sstem of linear equations b substitution. Check our solution. (Section 5.). = 5. + = = 3 + = 8 = 5 + = 0 Solve the sstem of linear equations b elimination. Check our solution. (Section 5.3) 7. + = = 9. 3 = 5 3 = = 5 + = 6 Solve the sstem of linear equations. (Section 5.) 0. =. 6 + = = = 6 = = 3. You plant a spruce tree that grows inches per ear and a hemlock tree that grows 6 inches per ear. The initial heights are shown. (Section 5.) a. Write a sstem of linear equations that represents this situation. b. Solve the sstem b graphing. Interpret our solution. in.. It takes ou 3 hours to drive to a concert 35 miles awa. You drive 55 miles per hour on highwas and 0 miles per hour on the rest of the roads. (Section 5., Section 5., and Section 5.3) a. How much time do ou spend driving at each speed? b. How man miles do ou drive on highwas? the rest of the roads? spruce tree hemlock tree 8 in. 5. In a football game, all of the home team s points are from 7-point touchdowns and 3-point field goals. The team scores si times. Write and solve a sstem of linear equations to find the numbers of touchdowns and field goals that the home team scores. (Section 5., Section 5., and Section 5.3) Chapter 5 Solving Sstems of Linear Equations