Systems of Linear Equations In Lesson 2, you gained experience in writing linear equations

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1 LESSON 3 Systems of Linear Equations In Lesson 2, you gained experience in writing linear equations with two variables to express a variety of problem conditions. Sometimes, problems involve two linear equations that have to be solved simultaneously. The task is to find one pair (x, y) of values that satisfies both linear equations. Students in the Hamilton High School science club faced this kind of problem when they tried to raise $240 to buy a special eyepiece for the high-powered telescope at their school. The school PTA offered to pay club members for an after-school work project that would clean up a nearby park and recreation center building. Because the outdoor work was harder and dirtier, the deal with the PTA would pay $16 for each outdoor worker and $10 for each indoor worker. The club had 18 members eager to work on the project. But, most members would prefer the easier indoor work. LESSON 3 Systems of Linear Equations UNIT 1 Functions, Equations, and Systems

2 Think About This Situation In Lesson 2, you learned that conditions in the science club s situation could be represented by linear equations like these: 16x + 10y = 240 and x + y = 18 a What do the variables x and y represent in these equations? b What problem condition is represented by each equation? c What are some combinations of numbers of outdoor and indoor workers that will allow the club to earn just enough money to buy the telescope eyepiece? Will any of those combinations also put each willing club member to work? d What different strategies could you use to find a pair of values for x and y that satisfy both linear equations simultaneously? Work on the problems of this lesson will develop your skill in writing, interpreting, and solving systems of linear equations. Investigation 1 Solving With Graphs and Substitution There are several different methods for solving systems of linear equations. As you work on the problems of this investigation, look for answers to this question: How can graphs and algebraic substitution be used to solve systems of linear equations? As you discussed in the Think About This Situation, a system of linear equations expressing the conditions in the science club s situation is: 16x + 10y = 240 x + y = 18 The first equation shows that the amount of money they can earn is a linear function of the variables x and y, where x and y represent the number of outdoor and indoor workers, respectively. The second equation shows that the number of club members who will work is also a linear function of those variables. 50 UNIT 1 Functions, Equations, and Systems 50 UNIT 1 Functions, Equations, and Systems

3 1 In Lesson 2, you learned that equations in the form ax + by = c are called linear equations because graphs of their solutions are straight lines. The diagram below shows graphs (in the first quadrant) of solutions to the equations 16x + 10y = 240 and x + y = y 20 Line Line x a. Match the graphs to the linear equations they represent. Explain how you know that your answers are correct. b. Use the graphs to estimate a solution (x, y) for the system of equations values for x and y that satisfy both equations. Explain what the solution tells about the science club s fund-raising situation. c. Since graphs give only estimates for solutions of equations, it is important to check the estimates. Show how your graph-based estimate can be checked to see if it is an exact solution to the system. 2 When the date for the work project was set, it turned out that only 13 science club members could participate. The club president talked again with the PTA president and got a new pay deal $20 per outdoor worker and $15 per indoor worker. a. Write a system of linear equations in which one equation expresses the new conditions about payment and the other shows the new number of workers. b. Estimate the solution for this system of equations by using graphs of the two equations. Then check your estimate. There are many cases when it is easy to solve systems of equations without taking the trouble to produce graphs and estimate the required x and y values. One such symbolic solution strategy, the substitution method, combines two equations with two variables into a single equation with only one variable by substituting from one equation into the other. LESSON 3 Systems of Linear Equations UNIT 1 Functions, Equations, and Systems

4 3 Recall the science club s goal of purchasing a $240 telescope eyepiece and the prospect of having 18 workers. Using the substitution method, you could start toward a solution of the system of equations representing this situation by reasoning like this. Step 1: We have to find values of x and y that satisfy both 16x + 10y = 240 and x + y = 18. Step 2: If x + y = 18, then y = 18 - x. Step 3: That means that 16x + 10(18 - x) = 240. This reasoning has led to an equation in Step 3 that contains only one variable, x. a. Explain each step in the reasoning. b. Continue the reasoning above to complete a solution of the system. Step 4: Solve the equation in Step 3 for x. What does this value of x tell you? Step 5: Use the value of x to find a value for y that satisfies the original equation. c. How could you check your solution from Part b? d. Explain what your solution for the system of equations tells about the numbers of club members doing outdoor and indoor work necessary to reach their earning goal of $ Look back at the reasoning used in Problem 3 to solve the system 16x + 10y = 240 and x + y = 18. a. In Step 2, the equation x + y = 18 can also be written as x = 18 - y. How can this fact be used to write an alternative to Step 3 that gives a single linear equation involving y alone? b. Check that this alternative strategy leads to the same solution for the original system. 5 Use reasoning similar to that in Problem 3 or 4 to solve the system of equations that you developed in Problem 2 to model the new conditions only 13 workers and different pay for each kind of work. 6 Use a graphing method to estimate solutions for three of the following systems of equations. Then use the substitution method to solve the three remaining systems. Check your solutions for all six systems by showing that the values of x and y you find make both equations true statements. a. y = 5x 6x - 2y = -4 b. 5x - y = -15 x + y = -3 c. 4x - y = 5 x = 8-2y d. -7x + y = 32 2x + 3y = 27 e. 2x + y = 5 4x - 3y = -10 f. 4x + 2y = 7 x - 5y = UNIT 1 Functions, Equations, and Systems 52 UNIT 1 Functions, Equations, and Systems

5 Summarize the Mathematics Many problem situations can be modeled by systems of linear equations. It is important to know how to solve such systems. a What does it mean to solve a system of linear equations? b How can you use graphs to estimate the solution for a system of linear equations? c Explain how to use the substitution method to solve a system of linear equations. d In what cases would graphing probably be more convenient than substitution? When would substitution be more convenient than graphing? e Once you have solved a system of equations, how can you check your solution? Be prepared to share your ideas with others in the class. Check Your Understanding As you complete the following tasks, think about the advantages and disadvantages of solving systems of linear equations by graphing and by substitution. a. Solve the following system of equations in two ways: by graphing the equations and by using substitution. 2x - y = 2 x + 2y = 18.5 b. The Kesling Middle School booster club is planning a community event to raise money for the school s art department. Based on previous fund-raising events, they estimate that the event will be a sellout filling all 300 seats in the school auditorium. Plans are to charge adults $8 and children $3 admission. The club wants to earn $2,000 from admission charges. i. Write an equation that expresses the relationship among adult attendance, child attendance, and the goal for income from admission charges. Explain what the variables represent. ii. Write an equation that expresses the relationship among number of adults, number of children, and total attendance. Explain what the variables represent. iii. Describe at least two different ways you could find the numbers of adults and children at the event if the club is to meet their income goal of $2,000 and their attendance estimate of 300 people. iv. Solve the system of linear equations and check your solution. LESSON 3 Systems of Linear Equations UNIT 1 Functions, Equations, and Systems