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SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS. ELIMINATION. WARM UP. How many solutions does this system have? x = - 3 and y + x = -2 Using your calculator, find the point of intersection of the following system. y = -3x + 6 and 2y – 4x = 2 Solve using substitution. y = 3x and x + 2y = -21

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SYSTEMS OF EQUATIONS

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  1. SYSTEMS OF EQUATIONS ELIMINATION

  2. WARM UP How many solutions does this system have? x = - 3 and y + x = -2 Using your calculator, find the point of intersection of the following system. y = -3x + 6 and 2y – 4x = 2 Solve using substitution. y = 3x and x + 2y = -21 Add: (3x – 5y + 16) and (2x + 5y – 31)

  3. Today we are going to look at solving a system of equations using elimination. What does the word elimination mean? Before we begin, write down how you think we are going to use elimination to solve a system. Discuss it with your partner. Be ready to share your thoughts.

  4. When we use the elimination method, we are using the properties of adding and subtracting equations in order to eliminate a variable in the system. Remember, we need to solve for one of the variables before we can find the value of the other variable. Look at the following examples. Try to identify which of the variables will be eliminated from each system.

  5. x – y = 12 2a + 4b = 30 x + y = 20 -2a – 2b = - 21.5 6x + 2y = - 10 - 4m + 2n = 6 2x + 2y = - 10 -4m + n = 8 2a – 3b = - 11 3m – 2n = 13 a + 3b = 8 m + 2n = 7 4x + y = - 9 3x – 5y = 16 4x + 2y = - 10 - 3x + 2y = - 10

  6. EXAMPLE ONE: SOLVING A SYSTEM BY ADDING EQUATIONS 2x + 5y = 17 and 6x – 5y = - 9 Step one: Eliminate one variable. Since the sum of the coefficients of y = 0, add the equations to eliminate y. 2x + 5y = 17 6x – 5y = -9 + 8x + 0 = 8 Add the two equations x = 1 Solve for x Step two: Substitute 1 for x to solve for the eliminated variable. 2x + 5y = 17 You can use any equation. 2(1) + 5y = 17 Substitute 1 for x. 2 + 5y = 17 Solve the equation. y = 3 Solution: (1, 3)

  7. Got it? Let’s try this one together. What is the solution to the system? Use elimination. 5x – 6y = - 32 and 3x + 6y = 48

  8. Now, try this one independently. When you are done, share your answer with your partner. - 3x – 3y = 9 and 3x – 4y = 5

  9. EXAMPLE TWO: SOLVING A SYSTEM BY SUBTRACTING EQUATIONS 5x + 2y = 6 and 9x + 2y = 22 Step One: Eliminate one variable. Since the difference of the coefficients of y = 0, eliminate y. 5x + 2y = 6 9x + 2y = 22 (-) Change all the signs by x( -1) - 4x = -16 x = 4 Step two: Solve for the eliminated variable. 5x + 2y = 6 Choose either equation. 5(4) + 2y = 6 Substitute for x. 20 + 2y = 6 Solve for y. 2y = 6 - 20 y = - 7 Solution: (4, - 7)

  10. Got it? Let’s try this one together. What is the solution to the system? Use elimination. x – 3y = 7 and x + 2y = 2

  11. Now, try this one independently. When you are done, share your answer with your partner. 3a + b = 5 and 2a + b = 10

  12. Sometimes neither variable in the system can be eliminated by simply adding or subtracting the equations. You can, however, use the Multiplication Property of Equality so that adding or subtracting eliminates one of the variables.

  13. EXAMPLE THREE: SOLVING A SYSTEM BY MULTIPLICATO]ION 3x + 4y = 6 and 5x + 2y = - 4 Step one: multiply the second equation by – 2 so the coefficients of the y terms are additive inverses. Then add. 3x + 4y = 6 X -2)- 10x – 4y = 8 Multiply through by – 2. -7x = 14 Add the equations. x = - 2 Solve for x. Step two: now substitute – 2 for x in either equation to find the value of y . 3x + 4y = 6 3(-2) + 4y = 6 Substitute in (-2). - 6 + 4y = 6 Simplify the equation. 4y = 12 Solve for y. y = 3 Solution: ( - 2, 3)

  14. Got it? Got it? Let’s try this one together. What is the solution to the system? Use elimination. -2x + 15y = - 32 and 7x – 5y = 17 Let’s try this one together. What is the solution to the system? Use elimination. -2x + 15y = - 32 and 7x – 5y = 17

  15. Now, try this one independently. When you are done, share your answer with your partner. -5x – 2y = 4 and 3x + 6y = 6

  16. EXAMPLE FOUR: SOLVING A SYSTEM BY MULTIPLICATO]ION BOTH EQUATIONS 3x + 2y = 1 and 4x + 3y = - 2 Step one: Multiply each equation so you can eliminate one variable. 3x + 2y = 1 multiply by 3 9x + 6y = 3 4x + 3y = - 2 multiply by -2 -8x - 6y = 4 add x = 7 Step two: Solve for the eliminated variable. Use either of the original equations. 3(7) + 2y = 1 Substitute in. 21 + 2y = 1 Simplify. 2y = - 20 Solve for y. y = - 10 Solution: ( 7, - 10)

  17. Got it? Let’s try this one together. What is the solution to the system? Use elimination. 3x – 2y = 1 and 8x + 3y = 2

  18. Now, try this one independently. When you are done, share your answer with your partner. 3p + q = 7 and 2p – 2q = -6

  19. Try the following examples. You may work with your group or partner. You must use elimination. Decide if you are going to add or subtract to solve. x + y = - 3 and x – y = 1 5. 3x – 7y = 6 and 2x + 7y = 4 8a + b = 1 and 8a – 3b = 3 6. 4x + 7y = 6 and 6x + 5y = 20 2m – 5n = - 6 and 2m – 7n = - 14 7. 4x – 3y = 12 and x + 2y = 14 4x – 3y = 12 and 4x + 3y = 24 8. 5x – 2y = - 15 and 3x + 8y = 37

  20. TICKET OUT THE DOOR Write a few sentences to summarize the lesson. Discuss the different ways to use elimination. How do you determine which method of elimination to use. Answer: If I had one question, what would it be?

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