Solve Systems By Elimination (Multiply First)

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Solve Systems By Elimination (Multiply First)

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  1. Objective

    Slide 1 - Objective

    • The student will be able to:
    • solve systems of equations using elimination with multiplication.
    • SOL: A.4e
    • Designed by Skip Tyler, Varina High School
  2. Solving Systems of Equations

    Slide 2 - Solving Systems of Equations

    • These notes go one step further and show how to use ELIMINATION with multiplication.
    • What happens when the coefficients are not the same?
    • We multiply the equations to make them the same! You’ll see…
  3. Slide 3

    • Solving a system of equations by elimination using multiplication.
    • Step 1: Put the equations in Standard Form.
    • Step 2: Determine which variable to eliminate.
    • Step 3: Multiply the equations and solve.
    • Step 4: Plug back in to find the other variable.
    • Step 5: Check your solution.
    • Standard Form: Ax + By = C
    • Look for variables that have the
    • same coefficient.
    • Solve for the variable.
    • Substitute the value of the variable
    • into the equation.
    • Substitute your ordered pair into
    • BOTH equations.
  4. 1) Solve the system using elimination.

    Slide 4 - 1) Solve the system using elimination.

    • 2x + 2y = 6
    • 3x – y = 5
    • Step 1: Put the equations in Standard Form.
    • Step 2: Determine which variable to eliminate.
    • They already are!
    • None of the coefficients are the same!
    • Find the least common multiple of each variable.
    • LCM = 6x, LCM = 2y
    • Which is easier to obtain?
    • 2y(you only have to multiplythe bottom equation by 2)
  5. 1) Solve the system using elimination.

    Slide 5 - 1) Solve the system using elimination.

    • Step 4: Plug back in to find the other variable.
    • 2(2) + 2y = 6
    • 4 + 2y = 6
    • 2y = 2
    • y = 1
    • 2x + 2y = 6
    • 3x – y = 5
    • Step 3: Multiply the equations and solve.
    • Multiply the bottom equation by 2
    • 2x + 2y = 6
    • (2)(3x – y = 5)
    • 8x = 16
    • x = 2
    • 2x + 2y = 6
    • (+) 6x – 2y = 10
  6. 1) Solve the system using elimination.

    Slide 6 - 1) Solve the system using elimination.

    • Step 5: Check your solution.
    • (2, 1)
    • 2(2) + 2(1) = 6
    • 3(2) - (1) = 5
    • 2x + 2y = 6
    • 3x – y = 5
    • Solving with multiplication adds one more step to the elimination process.
  7. 2) Solve the system using elimination.

    Slide 7 - 2) Solve the system using elimination.

    • x + 4y = 7
    • 4x – 3y = 9
    • Step 1: Put the equations in Standard Form.
    • They already are!
    • Step 2: Determine which variable to eliminate.
    • Find the least common multiple of each variable.
    • LCM = 4x, LCM = 12y
    • Which is easier to obtain?
    • 4x(you only have to multiplythe top equation by -4 to make them inverses)
  8. 2) Solve the system using elimination.

    Slide 8 - 2) Solve the system using elimination.

    • x + 4y = 7
    • 4x – 3y = 9
    • Step 4: Plug back in to find the other variable.
    • x + 4(1) = 7
    • x + 4 = 7
    • x = 3
    • Step 3: Multiply the equations and solve.
    • Multiply the top equation by -4
    • (-4)(x + 4y = 7)
    • 4x – 3y = 9)
    • y = 1
    • -4x – 16y = -28
    • (+) 4x – 3y = 9
    • -19y = -19
  9. 2) Solve the system using elimination.

    Slide 9 - 2) Solve the system using elimination.

    • Step 5: Check your solution.
    • (3, 1)
    • (3) + 4(1) = 7
    • 4(3) - 3(1) = 9
    • x + 4y = 7
    • 4x – 3y = 9
  10. What is the first step when solving with elimination?

    Slide 10 - What is the first step when solving with elimination?

    • Add or subtract the equations.
    • Multiply the equations.
    • Plug numbers into the equation.
    • Solve for a variable.
    • Check your answer.
    • Determine which variable to eliminate.
    • Put the equations in standard form.
  11. Which variable is easier to eliminate?

    Slide 11 - Which variable is easier to eliminate?

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    • 3x + y = 4
    • 4x + 4y = 6
    • x
    • y
    • 6
    • 4
  12. 3) Solve the system using elimination.

    Slide 12 - 3) Solve the system using elimination.

    • 3x + 4y = -1
    • 4x – 3y = 7
    • Step 1: Put the equations in Standard Form.
    • They already are!
    • Step 2: Determine which variable to eliminate.
    • Find the least common multiple of each variable.
    • LCM = 12x, LCM = 12y
    • Which is easier to obtain?
    • Either! I’ll pick y because the signs are already opposite.
  13. 3) Solve the system using elimination.

    Slide 13 - 3) Solve the system using elimination.

    • 3x + 4y = -1
    • 4x – 3y = 7
    • Step 4: Plug back in to find the other variable.
    • 3(1) + 4y = -1
    • 3 + 4y = -1
    • 4y = -4
    • y = -1
    • Step 3: Multiply the equations and solve.
    • Multiply both equations
    • (3)(3x + 4y = -1)
    • (4)(4x – 3y = 7)
    • x = 1
    • 9x + 12y = -3
    • (+) 16x – 12y = 28
    • 25x = 25
  14. 3) Solve the system using elimination.

    Slide 14 - 3) Solve the system using elimination.

    • Step 5: Check your solution.
    • (1, -1)
    • 3(1) + 4(-1) = -1
    • 4(1) - 3(-1) = 7
    • 3x + 4y = -1
    • 4x – 3y = 7
  15. What is the best number to multiply the top equation by to eliminate the x’s?

    Slide 15 - What is the best number to multiply the top equation by to eliminate the x’s?

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    • 3x + y = 4
    • 6x + 4y = 6
    • -4
    • -2
    • 2
    • 4
  16. Solve using elimination.

    Slide 16 - Solve using elimination.

    • 2x – 3y = 1
    • x + 2y = -3
    • (2, 1)
    • (1, -2)
    • (5, 3)
    • (-1, -1)