5.1 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 $6 preparing a bedroom to rent. The cost to our famil for food and utilities is $15 per night. The charge $75 per night to rent the bedroom. a. Write an equation that represents the costs. Cost, C (in dollars) = $15 per night Number of nights, + $6 b. Write an equation that represents the revenue (income). MODELING WITH 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 1 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) 1 5 6 7 8 9 1 11 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 1. 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. 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. =. 1. b. = c. = 1 = 1.7 +.7 = + 8 = + 5 Section 5.1 Solving Sstems of Linear Equations b Graphing 17
5.1 Lesson What You Will Learn Core Vocabular sstem of linear equations, p. 18 solution of a sstem of linear equations, p. 18 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 1 = 11 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. READING A sstem of linear equations is also called a linear sstem. a. (, 5); + = 7 Equation 1 = 11 Equation SOLUTION a. Substitute for and 5 for in each equation. Equation 1 Equation + = 7 b. (, ); = Equation 1 = + Equation = 11 + 5 =? 7 () (5) =? 11 7 = 7 11 = 11 Because the ordered pair (, 5) is a solution of each equation, it is a solution of the linear sstem. b. Substitute for and for in each equation. Equation 1 Equation = = + =? ( ) =? + = The ordered pair (, ) is a solution of the first equation, but it is not a solution of the second equation. So, (, ) 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. 1. (1, ); + = + = 5. (1, ); = + 1 = + 5 18 Chapter 5 Solving Sstems of Linear Equations
REMEMBER Note that the linear equations are in slope-intercept form. You can use the method presented in Section.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 1 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. Solving a Sstem of Linear Equations b Graphing Solve the sstem of linear equations b graphing. = + 5 Equation 1 = 1 Equation SOLUTION Step 1 Graph each equation. Step Estimate the point of intersection. The graphs appear to intersect at (1, ). Step Check our point from Step. Equation 1 Equation = + 5 = 1 (1, ) = + 5 = 1 =? (1) + 5 =? (1) 1 = = 1 The solution is (1, ). Check Use the table or intersect feature of a graphing calculator to check our answer. When = 1, the corresponding -values are equal. X - -1 1 X=1 Y1 9 7 5 1-1 - Y -9-5 -1 7 11 15 6 = + 5 6 Intersection X=1 Y= = 1 6 Monitoring Progress Solve the sstem of linear equations b graphing. Help in English and Spanish at BigIdeasMath.com. =. = 1 + 5. + = 5 = + = 5 = Section 5.1 Solving Sstems of Linear Equations b Graphing 19
Solving Real-Life Problems Modeling with Mathematics A roofing contractor bus bundles of shingles and rolls of roofing paper for $1. 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 1. 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.. Solve the Problem Words Price per bundle + Price per roll = 1 8 Price per bundle + Price per roll = 56 Variables Let be the price (in dollars) per bundle and let be the price (in dollars) per roll. Sstem + = 1 Equation 1 8 = 56 Equation Step 1 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 (, ). 16 = 7.5 + 6 = Step Check our point from Step. Equation 1 Equation + = 1 8 = 56 8 (, ) 8 16 () + () =? 1 8() =? 56 1 = 1 56 = 56 The solution is (, ). So, the price per bundle of shingles is $, and the price per roll of roofing paper is $.. Look Back You can use estimation to check that our solution is reasonable. A bundle of shingles costs about $. So, bundles of shingles and rolls of roofing paper (at $ per roll) cost about () + () = $98, and 8 bundles of shingles costs about 8() = $. These prices are close to the given values, so the solution seems reasonable. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 6. You have a total of 18 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
5.1 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY Do the equations 5 = 18 and 6 = 1 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 8, tell whether the ordered pair is a solution of the sstem of linear equations. (See Eample 1.). (, 6); + = 8 = 5. ( 1, ); = 7 = 8 + 5 6. (, ); = + 6 = 1 6 + 5 = 7 7. (, 1); = 8. (8, ); = 6 1 = 8. (5, 6); 6 + = 1 + = 1 In Eercises 9 1, use the graph to solve the sstem of linear equations. Check our solution. 9. = 1. + = 5 + = 1 = 1 In Eercises 1, solve the sstem of linear equations b graphing. (See Eample.) 1. = + 7 1. = + = + 1 = 8 15. = 1 + 16. = = + 5 = 1 + 11 17. 9 + = 18. = = = 5 19. =. + = = 1 + = 6 ERROR ANALYSIS In Eercises 1 and, describe and correct the error in solving the sstem of linear equations. 1. 1 The solution of the linear sstem = 6 and = is (, 1). 11. 6 + = 18 1. = + = + = 8 6. The solution of the linear sstem = 1 and = + 1 is =. Section 5.1 Solving Sstems of Linear Equations b Graphing 1
USING TOOLS In Eercises 6, use a graphing calculator to solve the sstem of linear equations... +. =. 1.6. =.6 +.6 =.6 +.6 = 6 5. 7 + 6 = 6. = 1.5.5 + = + = 1.5 7. MODELING WITH MATHEMATICS You have minutes to eercise at the gm, and ou want to burn calories total using both machines. How much time should ou spend on each machine? (See Eample.) Elliptical Trainer 8 calories per minute 8. MODELING WITH MATHEMATICS You sell small and large candles at a craft fair. You collect $1 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. 1. COMPARING METHODS Consider the equation + =. a. Solve the equation using algebra. b. Solve the sstem of linear equations = + and = 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)?. 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) 15 15 1 75 5 Buing Binders Compan A Compan B Compan C 15 5 5 5 5 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?. 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 miles from the trailhead and walks miles per hour. ou ( ) cm 6 cm. THOUGHT PROVOKING Your friend s bank account balance (in dollars) is represented b the equation = 5 + 5, 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 1.5). 1 + 5 = 5 + 5. 9 + 18 = 6 6. Chapter 5 Solving Sstems of Linear Equations Reviewing what ou learned in previous grades and lessons + 1 = 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.