Can a system of linear equations have no solution? Can a system of linear equations have many solutions?

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1 5. Solving Special Sstems of Linear Equations Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? ACTIVITY: Writing a Sstem of Linear Equations Work with a partner. Your cousin is ears older than ou. You can represent our ages b two linear equations. = t = t + Your age Your cousin s age a. Graph both equations in the same coordinate plane. b. What is the vertical distance between the two graphs? What does this distance represent? c. Do the two graphs intersect? Eplain what this means in terms of our age and our cousin s age t ACTIVITY: Using a Table to Solve a Sstem Work with a partner. You invest $500 for equipment to make dog backpacks. Each backpack costs ou $5 for materials. You sell each backpack for $5. Sstems of Equations In this lesson, ou will solve sstems of linear equations with no solution or infinitel man solutions. a. Cop and complete the table for our cost C and our revenue R C R b. When will ou break even? What is wrong? Chapter 5 Sstems of Linear Equations

2 Math Practice Analze Relationships What do ou know about the graphs of the two equations? How does this relate to the number of solutions? ACTIVITY: Using a Graph to Solve a Puzzle Work with a partner. Let and be two numbers. Here are two clues about the values of and. Clue : Clue : Words is more than twice the value of. The difference of and 6 is. Equation = + 6 = a. Graph both equations in the same coordinate plane. b. Do the two lines intersect? Eplain c. What is the solution of the puzzle? d. Use the equation = + to complete the table e. Does each solution in the table satisf both clues? f. What can ou conclude? How man solutions does the puzzle have? How can ou describe them?. IN YOUR OWN WORDS Can a sstem of linear equations have no solution? Can a sstem of linear equations have man solutions? Give eamples to support our answers. Use what ou learned about special sstems of linear equations to complete Eercises and on page 8. Section 5. Solving Special Sstems of Linear Equations 5

3 5. Lesson Lesson Tutorials Solutions of Sstems of Linear Equations A sstem of linear equations can have one solution, no solution, or infinitel man solutions. O O O One solution No solution Infinitel man solutions The lines intersect. The lines are parallel. The lines are the same. EXAMPLE Solving a Sstem: No Solution Solve the sstem. = + Equation = Equation Stud Tip You can solve some linear sstems b inspection. In Eample, notice ou can rewrite the sstem as + = + =. This sstem has no solution because + cannot be equal to and at the same time. 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. So, the sstem of linear equations has no solution. Method : Solve b substitution. Substitute for in Equation. = + Equation = + Substitute for. = Subtract from each side. The equation = is never true. So, the sstem of linear equations has no solution. 5 6 Eercises 8 0 Solve the sstem of linear equations. Check our solution.. = +. = 5. = + 0 = = 0 + = 6 Chapter 5 Sstems of Linear Equations

4 EXAMPLE Solving a Sstem: Infinitel Man Solutions Rectangle A Rectangle B The perimeter of Rectangle A is 6 units. The perimeter of Rectangle B is 08 units. Write and solve a sstem of linear equations to find the values of and. Perimeter of Rectangle A Perimeter of Rectangle B () + () = 6 (6) + () = = 6 Equation + = 08 Equation The sstem is: + 8 = 6 Equation + = 08 Equation 6 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. So, the sstem of linear equations has infinitel man solutions Method : Solve b elimination. Multipl Equation b and subtract the equations. + 8 = 6 Multipl b. + = 08 Revised Equation + = 08 + = 08 Equation 0 = 0 Subtract. The equation 0 = 0 is alwas true. In this contet, and must be positive. So, the solutions are all the points on the line + 8 = 6 in Quadrant I. The sstem of linear equations has infinitel man solutions. Eercises Solve the sstem of linear equations. Check our solution.. + = 5. + = 5 6. = 0 = + = 0 + = WHAT IF? What happens to the solution in Eample if the perimeter of Rectangle A is 5 units? Eplain. Section 5. Solving Special Sstems of Linear Equations 7

5 5. Eercises Help with Homework. WRITING Describe the difference between the graph of a sstem of linear equations that has no solution and the graph of a sstem of linear equations that has infinitel man solutions.. REASONING When solving a sstem of linear equations algebraicall, how do ou know when the sstem has no solution? infinitel man solutions? 9+(-6)= +(-)= +(-9)= 9+(-)= Let and be two numbers. Find the solution of the puzzle.. is more than times the value of.. of plus is equal to. The difference of and is. is 6 more than twice the value of. Without graphing, determine whether the sstem of linear equations has one solution, infinitel man solutions, or no solution. Eplain our reasoning. 5. = = = 8 = = + 8 = Solve the sstem of linear equations. Check our solution. 8. = 9. = + 0. = π + π = = π + = 6π. = =. + =. + 6 = = 6 (0.5 ) =.6. ERROR ANALYSIS Describe and correct re the error in solving the sstem of linear equations. = + = + 6 The lines have the same slope, so, there are infinitel man solutions. 5. PIG RACE In a pig race, our pig gets a head start of feet and is running at a rate of feet per second. Your friend s pig is also running at a rate of feet per second. A sstem of linear equations that represents this situation is = + and =. Will our friend s pig catch up to our pig? Eplain. 8 Chapter 5 Sstems of Linear Equations

6 6. REASONING One equation in a sstem of linear equations has a slope of. The other equation has a slope of. How man solutions does the sstem have? Eplain. 7. LOGIC How can ou use the slopes and the -intercepts of equations in a sstem of linear equations to determine whether the sstem has one solution, infinitel man solutions, or no solution? Eplain our reasoning. + 8 = = 8 8. MONEY You and a friend both work two different jobs. The sstem of linear equations represents the total earnings for hours worked at the first job and hours worked at the second job. Your friend earns twice as much as ou. a. One week, both of ou work hours at the first job. How man hours do ou and our friend work at the second job? b. Both of ou work the same number of hours at the second job. Compare the number of hours each of ou works at the first job. 9. DOWNLOADS You download a digital album for $0. Then ou and our friend download the same number of individual songs for $0.99 each. Write a sstem of linear equations that represents this situation. Will ou and our friend spend the same amount of mone? Eplain. 0. REASONING Does the sstem shown alwas, sometimes, or never have no solution when a = b? a b? a < b? Eplain our reasoning. = a + = b +. SKIING The table shows the number of lift tickets and ski rentals sold to two different groups. Is it possible to determine how much each lift ticket costs? Justif our answer.. Precision Find the values of a and b so the sstem shown has the solution (, ). Does the sstem have an other solutions? Eplain. Group Number of Lift Tickets 6 Number of Ski Rentals 8 Total Cost (dollars) b = a b = 6 Write an equation of the line that passes through the given points. (Section.7). (0, 0), (, 6). (0, ), (, ) 5. ( 6, 5), (0, ) 6. MULTIPLE CHOICE What is the solution of ( + 5) 6? (Skills Review Handbook) A B C D Section 5. Solving Special Sstems of Linear Equations 9

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