Solving Systems of Linear Equations by Elimination
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1 5.3 Solving Systems of Linear Equations by Elimination How can you use elimination to solve a system of linear equations? ACTIVITY: Using Elimination to Solve a System Work with a partner. Solve each system of linear equations by using two methods. Method : Subtract. Subtract Equation from Equation. What is the result? Explain how you can use the result to solve the system of equations. Method : Add. Add the two equations. What is the result? Explain how you can use the result to solve the system of equations. Is the solution the same using both methods? a. x + y = 4 b. 3x y = 4 c. x + y = 7 x y = 0 3x + y = x y = 5 ACTIVITY: Using Elimination to Solve a System Work with a partner. COMMON CORE Systems of Equations In this lesson, you will write and solve systems of linear equations by elimination. solve real-life problems. Learning Standards 8.EE.8b 8.EE.8c x + y = Equation x + 5y = Equation a. Can you add or subtract the equations to solve the system of linear equations? Explain. b. Explain what property you can apply to Equation in the system so that the y-coefficients are the same. c. Explain what property you can apply to Equation in the system so that the x-coefficients are the same. d. You solve the system in part (b). Your partner solves the system in part (c). Compare your solutions. e. Use a graphing calculator to check your solution. 6 Chapter 5 Systems of Linear Equations
2 3 ACTIVITY: Solving a Secret Code Math Practice Find Entry Points What is the first thing you do to solve a system of linear equations? Why? Work with a partner. Solve the puzzle to find the name of a famous mathematician who lived in Egypt around 350 A.D B W R M F Y K N O J A S I D X Z Q P C E G B T J M R C Z N O U W K X U H L Y S Q F E A S W K R M G J Z N H V D G 3 E L X L F Q O B x y x y 4 x y 3 x y 0 x y 4 x y x y 5 x y 5 x y 0 x y 3 3x 3y 0 x y 8 x y 5 x y 4. IN YOUR OWN WORDS How can you use elimination to solve a system of linear equations? 5. STRUCTURE When can you add or subtract equations in a system to solve the system? When do you have to multiply first? Justify your answers with examples. 6. LOGIC In Activity, why can you multiply equations in the system by a constant and not change the solution of the system? Explain your reasoning. Use what you learned about systems of linear equations to complete Exercises 4 6 on page. Section 5.3 Solving Systems of Linear Equations by Elimination 7
3 5.3 Lesson Lesson Tutorials Solving a System of Linear Equations by Elimination Step : Multiply, if necessary, one or both equations by a constant so at least pair of like terms has the same or opposite coefficients. Step : Add or subtract the equations to eliminate one of the variables. Step 3: Solve the resulting equation for the remaining variable. Step 4: Substitute the value from Step 3 into one of the original equations and solve. Study Tip Check Equation x + 3y = 7 + 3( 3) =? = Equation EXAMPLE Because the coefficients of x are the same, you can also solve the system by subtracting in Step. x + 3y = x 3y = 6 6y = 8 So, y = 3. x 3y = 6 7 3( 3) =? 6 6 = 6 Solving a System of Linear Equations by Elimination Solve the system by elimination. x + 3y = Equation x 3y = 6 Equation Step : The coefficients of the y-terms are already opposites. Step : Add the equations. x + 3y = Equation x 3y = 6 Equation x = 4 Add the equations. Step 3: Solve for x. x = 4 Equation from Step x = 7 Divide each side by. Step 4: Substitute 7 for x in one of the original equations and solve for y. x + 3y = Equation 7 + 3y = Substitute 7 for x. 3y = 9 Subtract 7 from each side. y = 3 Divide each side by 3. The solution is (7, 3). Exercises 7 Solve the system of linear equations by elimination. Check your solution.. x y = 9. 5x + y = x + 4y = 6 4x + y = 5x + y = 7x + 4y = 4 8 Chapter 5 Systems of Linear Equations
4 EXAMPLE Solving a System of Linear Equations by Elimination Solve the system by elimination. 6x + 5y = 5 Equation x 4y = 4 Equation Step : Multiply Equation by 3. 6x + 5y = 5 6x + 5y = 5 Equation x 4y = 4 Multiply by 3. 6x y = 4 Revised Equation Study Tip In Example, notice that you can also multiply Equation by 3 and then add the equations. Step : Subtract the equations. 6x + 5y = 5 Equation 6x y = 4 Revised Equation Step 3: Solve for y. 7y = 7 Subtract the equations. 7y = 7 Equation from Step y = Divide each side by 7. Step 4: Substitute for y in one of the original equations and solve for x. x 4y = 4 Equation x 4( ) = 4 Substitute for y. x + 4 = 4 x = 0 Multiply. Subtract 4 from each side. x = 5 Divide each side by. The solution is ( 5, ). Check 0 6x 5y 5 x 4y Solve the system of linear equations by elimination. Check your solution. 4. 3x + y = 5. 4x 5y = y = 5 5x 6x + 3y = 4 x y = 8 y = x + 3 Section 5.3 Solving Systems of Linear Equations by Elimination 9
5 EXAMPLE 3 Real-Life Application You buy 8 hostas and 5 daylilies for $93. Your friend buys 3 hostas and daylilies for $7. Write and solve a system of linear equations to find the cost of each daylily. Use a verbal model to write a system of linear equations. Number of hostas Cost of each hosta, x + Number of daylilies Cost of each daylily, y = Total cost The system is: 8x + 5y = 93 Equation (You) 3x + y = 7 Equation (Your friend) Step : To find the cost y of each daylily, eliminate the x-terms. Multiply Equation by 3. Multiply Equation by 8. 8x + 5y = 93 Multiply by 3. 4x + 45y = 579 Revised Equation 3x + y = 7 Multiply by 8. 4x + 96y = 936 Revised Equation Step : Subtract the revised equations. 4x + 45y = 579 Revised Equation 4x + 96y = 936 Revised Equation 5y = 357 Subtract the equations. Step 3: Solving the equation 5y = 357 gives y = 7. So, each daylily costs $7. Exercises 6 7. A landscaper buys 4 peonies and 9 geraniums for $90. Another landscaper buys 5 peonies and 6 geraniums for $85. Write and solve a system of linear equations to find the cost of each peony. Methods for Solving Systems of Linear Equations Method Graphing (Lesson 5.) Substitution (Lesson 5.) Elimination (Lesson 5.3) Elimination (Multiply First) (Lesson 5.3) To estimate solutions When to Use When one of the variables in one of the equations has a coefficient of or When at least pair of like terms has the same or opposite coefficients When one of the variables cannot be eliminated by adding or subtracting the equations 0 Chapter 5 Systems of Linear Equations
6 5.3 Exercises Help with Homework. WRITING Describe how to solve a system of linear equations by elimination.. NUMBER SENSE When should you use multiplication to solve a system of linear equations by elimination? 3. WHICH ONE DOESN T BELONG? Which system of equations does not belong with the other three? Explain your reasoning. 3x + 3y = 3 x 3y = 7 x + y = 6 x 3y = 0 x + 3y = 3x y = 0 x + y = 5 3x y = 3 9+(-6)=3 3+(-3)= 4+(-9)= 9+(-)= Use a method from Activity to solve the system. 4. x + y = 3 5. x + 3y = x + y = 3 x y = x + 3y = 3x y = 9 Solve the system of linear equations by elimination. Check your solution. 7. x + 3y = 5 8. x y = x + 3y = 5 x y = 3 3x + y = 3 x + 3y = 0 0. x + 7y =. x + 5y = 6. 3x y = 4 x 4y = 3x 5y = 6x y = 3. ERROR ANALYSIS Describe and correct the error in solving the system of linear equations. 4. RAFFLE TICKETS You and your The solution is (5, 8). friend are selling raffle tickets for a new laptop. You sell 4 more tickets than your friend sells. Together, you and your friend sell 58 tickets. a. Write a system of linear equations that represents this situation. b. How many tickets does each of you sell? 5. JOGGING You can jog around your block twice and the park once in 0 minutes. You can jog around your block twice and the park 3 times in minutes. a. Write a system of linear equations that represents this situation. b. How long does it take you to jog around the park? 5x + y = 9 Equation 3x y = Equation x = 0 x = 5 Section 5.3 Solving Systems of Linear Equations by Elimination
7 3 Solve the system of linear equations by elimination. Check your solution. 6. x y = 0 7. x + 4y = 8. x + 3y = 7 3x y = 3 3x + 5y = 0 5x + 8y = 9. 3x + 3 = 3y 0. x 6 = 4y. 5x = 4y + 8 x 6y = 7y = 3x + 9 3y = 3x 3. ERROR ANALYSIS Describe and correct the error in solving the system of linear equations. x + y = Equation Multiply by 5. 5x + 5y = 5 5x + 3y = 3 Equation 5x + 3y = 3 8y = 8 y = The solution is (, ). 3. REASONING For what values of a and b should you solve the system by elimination? a. 4x y = 3 b. x 7y = 6 ax + 0y = 6 6x + by = 9 Determine whether the line through the first pair of points intersects the line through the second pair of points. Explain. 4. Line : (, ), (, 7) 5. Line : (3, ), (7, ) Line : ( 4, ), (0, 5) Line : (5, ), (6, ) 6. AIRPLANES Two airplanes are flying to the same airport. Their positions are shown in the graph. Write a system of linear equations that represents this situation. Solve the system by elimination to justify your answer. y Airport x 7. TEST PRACTICE The table shows the number of correct answers on a practice standardized test. You score 86 points on the test, and your friend scores 76 points. You Your Friend Multiple Choice 3 8 Short Response 0 5 a. Write a system of linear equations that represents this situation. b. How many points is each type of question worth? Chapter 5 Systems of Linear Equations
8 8. LOGIC You solve a system of equations in which x represents the number of adult tickets sold and y represents the number of student tickets sold. Can ( 6, 4) be the solution of the system? Explain your reasoning. 9. VACATION The table shows the activities of two tourists at a vacation resort. You want to go parasailing for hour and horseback riding for hours. How much do you expect to pay? Parasailing Horseback Riding Total Cost Tourist hours 5 hours $05 Tourist 3 hours 3 hours $ REASONING The solution of a system of linear equations is (, 4). One equation in the system is x + y = 0. Explain how you could find a second equation for the system. Then find a second equation. Solve the system by elimination to justify your answer. 3. JEWELER A metal alloy is a mixture of two or more metals. A jeweler wants to make 8 grams of 8-carat gold, which is 75% gold. The jeweler has an alloy that is 90% gold and an alloy that is 50% gold. How much of each alloy should the jeweler use? 3. PROBLEM SOLVING A powerboat takes 30 minutes to travel 0 miles downstream. The return trip takes 50 minutes. What is the speed of the current? 33. Solve the system of equations by elimination. x y + 3z = x + y 4z = y z = 0 Decide whether the two equations are equivalent. (Section. and Section.3) 34. 4n + = n a + 6 = 36. 7v 3 = 5 3n = 9 a + 3 = 6 4v 3 = MULTIPLE CHOICE Which line has the same slope as y = x 3? (Section 4.4) A y = x + 4 B y = x + 3 C y x = 5 D y x = 7 Section 5.3 Solving Systems of Linear Equations by Elimination 3
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