. Solving Sstems of Linear Equations b Graphing How can ou solve a sstem of linear equations? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your famil starts a bed-and-breakfast. The spend $5 fiing up a bedroom to rent. The cost for food and utilities is $ per night. Your famil charges $ per night to rent the bedroom. a. Write an equation that represents the costs. Cost, C (in dollars) = $ per night Number of nights, + $5 b. Write an equation that represents the revenue (income). Revenue, R (in dollars) = $ per night Number of 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. ACTIVITY: Using a Table to Solve a Sstem COMMON CORE Sstems of Equations In this lesson, ou will write and solve sstems of linear equations b graphing. solve real-life problems. Learning Standards.EE.a.EE.b.EE.c A.CED. A.REI. Use the cost and revenue equations from Activit to find how man nights our famil needs to rent the bedroom before recovering the cost of fiing up the bedroom. This is the break-even point. a. Cop and complete the table. 5 7 9 C R b. How man nights does our famil need to rent the bedroom before breaking even? 5 Chapter Solving Sstems of Linear Equations
ACTIVITY: Using a Graph to Solve a Sstem a. Graph the cost equation from Activit. b. In the same coordinate plane, graph the revenue equation from Activit. c. Find the point of intersection of the two graphs. What does this point represent? How does this compare to the break-even point in Activit? Eplain. 5 5 7 9 EXAMPLE: Using a Graphing Calculator Math Practice Use Technolog to Eplore How do ou decide the values for the viewing window of our calculator? What other viewing windows could ou use? Use a graphing calculator to solve the sstem. = + 5 Equation a. Enter the equations into our calculator. Then graph the equations in an appropriate window. = Equation b. To find the solution, use the intersect feature to find the point of intersection. The solution is (, ). 5 5. IN YOUR OWN WORDS How can ou solve a sstem of linear equations? How can ou check our solution?. Solve one of the sstems b using a table, another sstem b sketching a graph, and the remaining sstem b using a graphing calculator. Eplain wh ou chose each method. a. =. +. b. = c. = 5 =.7. = + 9 = + Use what ou learned about sstems of linear equations to complete Eercises on page 5. Section. Solving Sstems of Linear Equations b Graphing 55
. Lesson Lesson Tutorials Ke Vocabular sstem of linear equations, p. 5 solution of a sstem of linear equations, p. 5 A sstem of linear equations is a set of two or more linear equations in the same variables. An eample is shown below. = + Equation = 7 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. The solution of a sstem of linear equations is the point of intersection of the graphs of the equations. Reading A sstem of linear equations is also called a linear sstem. Solving a Sstem of Linear Equations b Graphing Step Graph each equation in the same coordinate plane. Step Estimate the point of intersection. Step Check the point from Step b substituting for and in each equation of the original sstem. EXAMPLE Solving a Sstem of Linear Equations b Graphing Solve the sstem b graphing. = + 5 Equation = Equation Check 5 Step : Graph each equation. Step : Estimate the point of intersection. The graphs appear to intersect at (, ). Step : Check the point from Step. Equation Equation = + 5 = =? ( ) + 5 =? ( ) = = 5 (, ) 5 5 5 The solution is (, ). Eercises Solve the sstem of linear equations b graphing.. =. = 5 +. = = + = = + 5 Chapter Solving Sstems of Linear Equations
EXAMPLE Real-Life Application A kicker on a football team scores point for making an etra point and points for making a field goal. The kicker makes a total of etra poi nts and field goals in a game and scores points. Write and solve a ss tem of linear equations to find the number of etra points and the number of field goals. Use a verbal model to write a sstem of linear equations. Number of etra points, + Number of field goals, = Total number of kicks Points per etra point Number of etra + points, Points per field goal Number of field goals, = Total number of points Stud Tip It ma be easier to graph the equations in a sstem b rewriting the equations in slope-intercept form. The sstem is: + = Equation Step : Graph each equation. + = Equation Step : Estimate the point of intersection. The graphs appear to intersect at (, ). Step : Check our point from Step. Equation Equation + = + = + =? + () =? = = The solution is (, ). So, the kicker made etra points and field goals. 7 5 Check (, ) 5 7 Eercises 5 Solve the sstem of linear equations b graphing.. = 7 5. = 5. + = + = + = + = 7. WHAT IF? In Eample, the kicker makes a total of 7 etra points and field goals and scores 7 points. Write and solve a sstem of linear equations to find the numbers of etra points and field goals. Section. Solving Sstems of Linear Equations b Graphing 57
. Eercises Help with Homework. VOCABULARY Do the equations = 5 and 7 + = form a sstem of linear equations? Eplain.. WRITING What does it mean to solve a sstem of equations?. WRITING You graph a sstem of linear equations and the solution appears to be (, ). How can ou verif that the solution is (, )? 9+(-)= +(-)= +(-9)= 9+(-)= Use a table to find the break-even point. Check our solution.. C = 5 + 5 5. C = +. C = + R = 5 R = R = 7 Match the sstem of linear equations with the corresponding graph. Use the graph to estimate the solution. Check our solution. 7. =.5. = + 9. = = + = = + 5 A. B. C. Solve the sstem of linear equations b graphing.. = + 9. =. = + 5 = = 5 + =. + = 7. = 7 5. = 7 = + = +.5 + = 5. CARRIAGE RIDES The cost C (in dollars) for the care and maintenance of a horse and carriage is C = 5 +, where is the number of rides. a. Write an equation for the revenue R in terms of the number of rides. b. How man rides are needed to break even? 5 Chapter Solving Sstems of Linear Equations
Use a graphing calculator to solve the sstem of linear equations. 7.. + =.5.. +. =.7 9.. 5.5 =.. = 5.7.9 =... =.. ERROR ANALYSIS Describe and correct the error in solving the sstem of linear equations.. REASONING Is it possible for a sstem of two linear equations to have eactl two solutions? Eplain our reasoning. 5 5 The solution of the linear sstem =.5 + and = + 7 is =.. MODELING You have a total of math and science problems for homework. You have more math problems than science problems. How man problems do ou have in each subject? Use a sstem of linear equations to justif our answer.. CANOE RACE You and our friend are in a canoe race. Your friend is a half mile in front of ou and paddling miles per hour. You are paddling. miles per hour. our friend ou a. You are.5 miles from the finish line. How long will it take ou to catch up to our friend? b. You both maintain our paddling rates for the remainder of the race. How far ahead of our friend will ou be when ou cross the finish line?. Your friend is tring to grow her hair as long as her cousin s hair. The table shows their hair lengths (in inches) in different months. Month Friend s Hair (in.) Cousin s Hair (in.) 7.5 9 a. Write a sstem of linear equations that represents this situation. b. Will our friend s hair ever be as long as her cousin s hair? If so, in what month? Solve the equation. Check our solution. (Section.) 5. c c + = 7. 5( ) + = 7. ( + ) = 9. MULTIPLE CHOICE The graph of which equation is perpendicular to the graph of = +? (Section.) A = B = + C = D = + Section. Solving Sstems of Linear Equations b Graphing 59