? LESSN. Solving Sstems of Linear Equations b Graphing ESSENTIAL QUESTIN How can ou solve a sstem of equations b graphing? Epressions, equations, and relationships.9 Identif and verif the values of and that simultaneousl satisf two linear equations in the form = m + b from the intersections of the graphed equations. EXPLRE ACTIVITY Investigating Sstems of Equations You have learned several was to graph a linear equation in slope-intercept form. For eample, ou can use the slope and -intercept or ou can find two points that satisf the equation and connect them with a line. A Graph the pair of equations together: { = 3 - = - + 3..9 Slope-intercept form is = m + b, where m is the slope and b is the -intercept. B Eplain how to tell whether (, -1) is a solution of the equation = 3 - without using the graph. - C Eplain how to tell whether (, -1) is a solution of the equation = - + 3 without using the graph. - D E Eplain how to use the graph to tell whether the ordered pair (, -1) is a solution of either equation. Find the point of intersection of the two lines. Check b substitution to determine if it is a solution to both equations. Point of intersection, = 3 - ( ) = - + 3 = 3-1 = = - + 3 1 = The point of intersection is / is not the solution of both equations. Lesson. 11
Math n the Spot m.hrw.com Solving Sstems Graphicall An ordered pair (, ) is a solution of an equation in two variables if substituting the - and -values into the equation results in a true statement. A sstem of equations is a set of equations that have the same variables. An ordered pair is a solution of a sstem of equations if it is a solution of ever equation in the set. Since the graph of an equation represents all ordered pairs that are solutions of the equation, if a point lies on the graphs of two equations, the point is a solution of both equations and is, therefore, a solution of the sstem. EXAMPLE 1.9 M Notes Solve each sstem b graphing. { = - + A = 3 STEP 1 Start b graphing each equation. STEP Find the point of intersection of the two lines. It appears to be (1, 3). Check b substitution to determine if it is a solution to both equations. - = - + = 3-3? = -(1) + 3? = 3(1) 3 = 3 3 = 3 B The solution of the sstem is (1, 3). { = 3-3 = 3( - 1) STEP 1 STEP Start b graphing each equation. Identif an ordered pairs that are solutions of both equations. The graphs of the equations are the same line. So, ever ordered pair that is a solution of one equation is also a solution of the other equation. The sstem has infinitel man solutions. - 11 Unit
Reflect 1. A sstem of linear equations has infinitel man solutions. Does that mean an ordered pair in the coordinate plane is a solution?. Can ou show algebraicall that both equations in part B represent the same line? If so, eplain how. YUR TURN Solve each sstem b graphing. Check b substitution. 3. { = - + = - - 1 Check: - - - - - -. { = - + = 3 Check: - - - - - - Personal Math Trainer nline Assessment and Intervention m.hrw.com Lesson. 117
Math n the Spot m.hrw.com Solving Problems Using Sstems of Equations When using graphs to solve a sstem of equations, it is best to rewrite both equations in slope-intercept form for ease of graphing. To write an equation in slope-intercept form starting from a + b = c: a + b = c b = c - a = c b - a b = - a b + c b Subtract a from both sides. Divide both sides b b. Rearrange the equation. EXAMPLE.9 Keisha and her friends visit the concession stand at a football game. The stand charges $ for a hot dog and $1 for a drink. The friends bu a total of items for $11. Tell how man hot dogs and how man drinks the bought. STEP 1 Let represent the number of hot dogs the bought and let represent the number of drinks the bought. Write an equation representing the number of items the purchased. Number of hot dogs + Number of drinks = Total items + = 11 Unit STEP Write an equation representing the mone spent on the items. Cost of 1 hot dog times + Cost of 1 drink times number of hot dogs number of drinks Drinks 10 = Total cost + 1 = 11 Write the equations in slope-intercept form. Then graph. + = = - = - + + 1 = 11 1 = 11 - = - + 11 Graph the equations = - + and = - + 11. Hot dogs 10 Image Credits: Weronica Ankorhorn/Houghton Mifflin Harcourt
STEP 3 STEP Use the graph to identif the solution of the sstem of equations. Check our answer b substituting the ordered pair into both equations. Apparent solution: (3, ) Check: + = + = 11 3 + =? (3) + =? 11 = 11 = 11 The point (3, ) is a solution of both equations. Interpret the solution in the original contet. Keisha and her friends bought 3 hot dogs and drinks. Animated Math m.hrw.com Reflect. Conjecture Wh do ou think the graph is limited to the first quadrant? YUR TURN. During school vacation, Marquis wants to go bowling and to pla laser tag. He wants to pla total games but needs to figure out how man of each he can pla if he spends eactl $0. Each game of bowling is $ and each game of laser tag is $. a. Let represent the number of games Marquis bowls and let represent the number of games of laser tag Marquis plas. Write a sstem of equations that describes the situation. Then write the equations in slope-intercept form. b. Graph the solutions of both equations. c. How man games of bowling and how man games of laser tag will Marquis pla? Games of laser tag 10 Games of bowling 10 Personal Math Trainer nline Assessment and Intervention m.hrw.com Lesson. 119
Guided Practice Solve each sstem b graphing. (Eample 1) 1. { = 3 - = +. { - 3 = -3 + 9 = - - - - - - - - - 3. Mrs. Morales wrote a test with 1 questions covering spelling and vocabular. Spelling questions () are worth points and vocabular questions () are worth 10 points. The maimum number of points possible on the test is 100. (Eample ) a. Write an equation in slope-intercept form to represent the number of questions on the test. 0 b. Write an equation in slope-intercept form to represent the total number of points on the test. Caps 1 1? c. Graph the solutions of both equations. d. Use our graph to tell how man of each question tpe are on the test. ESSENTIAL QUESTIN CHECK-IN. When ou graph a sstem of linear equations, wh does the intersection of the two lines represent the solution of the sstem? 1 1 0 Shirts 10 Unit
Name Class Date. Independent Practice.9. Vocabular A is a set of equations that have the same variables.. Eight friends started a business. The will wear either a baseball cap or a shirt imprinted with their logo while working. The want to spend eactl $3 on the shirts and caps. Shirts cost $ each and caps cost $3 each. a. Write a sstem of equations to describe the situation. Let represent the number of shirts and let represent the number of caps. m.hrw.com Personal Math Trainer nline Assessment and Intervention 7. Multistep The table shows the cost for bowling at two bowling alles. Shoe Rental Fee Cost per Game Bowl-o-Rama $.00 $.0 Bowling Pinz $.00 $.00 a. Write a sstem of equations, with one equation describing the cost to bowl at Bowl-o-Rama and the other describing the cost to bowl at Bowling Pinz. For each equation, let represent the number of games plaed and let represent the total cost. b. Graph the sstem to find the solution. Verif the solution. What does it represent? 0 1 b. Graph the sstem to find the solution. Verif the solution. What does it represent? Cost of Bowling Caps 1 1 1 0 Shirts Cost ($) 1 1 Games Lesson. 11
. Multi-Step Jerem runs 7 miles per week and increases his distance b 1 mile each week. Ton runs 3 miles per week and increases his distance b miles each week. In how man weeks will Jerem and Ton be running the same distance? What will that distance be? 9. Critical Thinking Write a real-world situation that could be represented b the sstem of equations shown below. = + 10 { = 3 + 1 FCUS N HIGHER RDER THINKING Work Area 10. Multistep The table shows two options provided b a high-speed Internet provider. Setup Fee ($) Cost per Month ($) ption 1 0 30 ption No setup fee $0 a. In how man months will the total cost of both options be the same? What will that cost be? b. If ou plan to cancel our Internet service after 9 months, which is the cheaper option? Eplain. 11. Draw Conclusions How man solutions does the sstem formed b - = 3 and a - a + 3a = 0 have for a nonzero number a? Eplain. 1 Unit