# 11.3 Solving Linear Systems by Adding or Subtracting

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1 Name Class Date 11.3 Solving Linear Systems by Adding or Subtracting Essential Question: How can you solve a system of linear equations by adding and subtracting? Resource Locker Explore Exploring the Effects of Adding Equations Systems of equations can be solved by graphing, substitution, or by a third method, called elimination. A Look at the system of linear equations. - 4y = -10 2x 3x + 4y = 5 What do you notice about the coefficients of the y-terms? B What is the sum of -4y and 4y? How do you know? C Find the sum of the two equations by combining like terms. D Use the equation from Step C to find the value of x. 2x -4y = x +4y = +5 + = x = E Use the value of x to find the value of y. What is the solution of the system? y = Solution: Reflect 1. Discussion How do you know that when both sides of the two equations were added, the resulting sums were equal? 2. Discussion How could you check your solution? Module Lesson 3

2 Explain 1 Solving Linear Systems by Adding or Subtracting The elimination method is a method used to solve systems of equations in which one variable is eliminated by adding or subtracting two equations in the system. Steps in the Elimination Method 1. Add or subtract the equations to eliminate one variable, and then solve for the other variable. 2. Substitute the value into either original equation to find the value of the eliminated variable. 3. Write the solution as an ordered pair. Example 1 Solve each system of linear equations using the indicated method. Check your answer by graphing. A Solve the system of linear equations by adding. 4x - 2y = 12 x + 2y = 8 Add the equations. 4x - 2y = 12 x + 2y = 8 5x + 0 = 20 5x = 20 x = 4 Substitute the value of x into one of the equations and solve for y. x + 2y = y = 8 2y = 4 y = 2 Write the solution as an ordered pair. (4, 2) 8 x + 2y = y x 4 8 4x - 2y = 12 B Check the solution by graphing. Solve the system of linear equations by subtracting. 2x + 6y = 6 Substitute the value of y into one of the equations and solve for x. 2x - y = -8 Subtract the equations. 2x + 6y = 6 -(2x - y = - 8) = Module Lesson 3

3 Write the solution as an ordered pair. 8 y -8 4 x Check the solution by graphing. Reflect 3. Can the system in part A be solved by subtracting one of the original equations from the other? Why or why not? 4. In part B, what would happen if you added the original equations instead of subtracting? Your Turn Solve each system of linear equations by adding or subtracting y = -24 2x 3x - 5y = y = 5 3x x + 2y = -1 Module Lesson 3

4 Explain 2 Solving Special Linear Systems by Adding or Subtracting Example 2 Solve each system of linear equations by adding or subtracting. -4x - 2y = 4 A 4x + 2y = -4 Add the equations. -4x - 2y = 4 +4x + 2y = = 0 0 = 0 The resulting equation is true, so the system has infinitely many solutions. Graph the equations to provide more information. 8 y 4 x The graphs are the same line, so the system has infinitely many solutions. x + y = -2 B x + y = 4 Subtract the equations. x + y = -2 - (x + y = 4) The resulting equation is, so the system has solutions. Graph the equations to provide more information. 8 y 4 x The graph shows that the lines are and. Module Lesson 3

5 Your Turn Solve each system of linear equations by adding or subtracting. 7. 4x - y = 3 4x - y = x - 6y = 7 -x + 6y = -7 Explain 3 Solving Linear System Models by Adding or Subtracting Example 3 Solve by adding or subtracting. A Perfect Patios is building a rectangular deck for a customer. According to the customer s specifications, the perimeter should be 40 meters and the difference between twice the length and twice the width should be 4 meters. The system of equations 2l + 2w = 40 can be used to 2l - 2w = 4 represent this situation, where l is the length and w is the width. What will be the length and width of the deck? Image Credits: Huntstock, Inc/Alamy Add the equations. 2l + 2w = 40 2l - 2w = 4 4l + 0 = 44 4l = 44 l = 11 Write the solution as an ordered pair. (l, w) = (11, 9) Substitute the value of l into one of the equations and solve for w. 2l + 2w = 40 2 (11) + 2w = w = 40 2w = 18 w = 9 The length of the deck will be 11 meters and the width will be 9 meters. Module Lesson 3

6 B A video game and movie rental kiosk charges \$2 for each video game rented, and \$1 for each movie rented. One day last week, a total of 114 video games and movies were rented for a total of \$177. The system of equations x + y = 114 represents this situation, where x 2x + y = 177 represents the number of video games rented and y represents the number of movies rented. Find the numbers of video games and movies that were rented. Subtract the equations. x + y = (2x + y = 177) Substitute the value of x into one of the equations and solve for y. Write the solution as an ordered pair. video games and movies were rented. Your Turn 9. The perimeter of a rectangular picture frame is 62 inches. The difference of the length of the frame and twice its width is 1. The system of equations + 2w = 62 2l represents this situation, l - 2w = 1 where l represents the length in inches and w represents the width in inches. What are the length and the width of the frame? Elaborate 10. How can you decide whether to add or subtract to eliminate a variable in a linear system? Explain your reasoning. 11. Discussion When a linear system has no solution, what happens when you try to solve the system by adding or subtracting? 12. Essential Question Check-In When you solve a system of linear equations by adding or subtracting, what needs to be true about the variable terms in the equations? Module Lesson 3

7 Evaluate: Homework and Practice 1. Which method of elimination would be best to solve the system of linear equations? Explain. 1_ 2 x + 3 _ 4 y = -10 -x - 3 _ 4 y = 1 Online Homework Hints and Help Extra Practice Solve each system of linear equations by adding or subtracting. 2. 3x + 2y = 10 3x - y = x + y = 3 3x - y = x + y = 5 x - 3y = x + y = -4 2x - y = x + y = -3 5x - 3y = x + y = -6-5x + y = x - 3y = 15 4x - 3y = x - 6y = 36-2x + 6y = 0 Module Lesson 3

8 10. 1_ 2 x - 7_ 9 y = - _ y = x - 1_ 2 x + 7_ 9 y = 6 2_ 3-10x + 2y = y = 7-2x 2x - 5y = x + y = 0 -x - y = y = -3-5x -5x - y = by = c ax ax - by = c 16. The sum of two numbers is 65, and the difference of the numbers is 27. The system of linear equations + y = 65 x represents this situation, where x is the larger number and y is the smaller x - y = 27 number. Solve the system to find the two numbers. Module Lesson 3

9 17. A rectangular garden has a perimeter of 120 feet. The length of the garden is 24 feet greater than twice the width. The system of linear equations + 2w = 120 2l l - 2w = 24 represents this situation, where l is the length of the garden and w is its width. Find the length and width of the garden. 18. The sum of two angles is 90. The difference of twice the larger angle and the smaller angle is 105. The system of linear equations x + y = 90 represents this 2x - y = 105 situation where x is the larger angle and y is the smaller angle. Find the measures of the two angles. 19. Max and Sasha exercise a total of 20 hours each week. Max exercises 15 hours less than 4 times the number of hours Sasha exercises. The system of equations x + y = 20 represents this x - 4y = -15 situation, where x represents the number of hours Max exercises and y represents the number of hours Sasha exercises. How many hours do Max and Sasha exercise per week? 20. The sum of the digits in a two-digit number is 12. The digit in the tens place is 2 more than the digit in the ones place. The system of linear equations + y = 12 x represents this situation, x - y = 2 where x is the digit in the tens place and y is the digit in the ones place. Solve the system to find the two-digit number. Module Lesson 3

10 21. A pool company is installing a rectangular pool for a new house. The perimeter of the pool must be 94 feet, and the length must be 2 feet more than twice the width. The system of linear equations + 2w = 94 2l represents l = 2w + 2 this situation, where l is the length and w is the width. What are the dimensions of the pool? 22. Use one solution, no solutions, or infinitely many solutions to complete each statement. a. When the solution of a system of linear equations yields the equation 4 = 4, the system has. b. When the solution of a system of linear equations yields the equation x = 4, the system has. c. When the solution of a system of linear equations yields the equation 0 = 4, the system has. H.O.T. Focus on Higher Order Thinking 23. Multiple Representations You can use subtraction to solve the system of linear equations shown. + 4y = -4 2x 2x - 2y = -10 Instead of subtracting 2x - 2y = -10 from 2x + 4y = -4, what equation can you add to get the same result? Explain. Image Credits: Marlene Ford/Alamy Module Lesson 3

11 24. Explain the Error Liang s solution of a system of linear equations is shown. Explain Liang s error and give the correct solution. 3x - 2y = 12 -x - 2y = -20 3x - 2y = 12 -x - 2y = -20 2x = -8 x = -4 3x - 2y = 12 3 (-4) - 2y = y = 12-2y = 24 y = -12 Solution: (-4, -12) 25. Represent Real-World Problems For a school play, Rico bought 3 adult tickets and 5 child tickets for a total of \$40. Sasha bought 1 adult ticket and 5 child tickets for a total of \$25. The system of linear equations + 5y = 40 3x x + 5y = 25 represents this Image Credits: aerogondo2/shutterstock situation, where x is the cost of an adult ticket and y is the cost of a child ticket. How much will Julia pay for 5 adult tickets and 3 child tickets? Module Lesson 3

12 Lesson Performance Task A local charity run has a Youth Race for runners under the age of 12. The entry fee is \$5 for an individual or \$4 each for two runners from the same family. Carter is collecting the registration forms and fees. After everyone has registered, he picks up the cash box and finds a dollar on the ground. He checks the cash box and finds that it contains \$200 and the registration slips for 47 runners. Does the dollar belong in the cash box or not? Explain your reasoning. (Hint: You can use the system of equations i + f = 47 and 5i + 4f = 200, where i equals the number of individual tickets and f equals the number of family tickets.) Module Lesson 3

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