Solve Systems By Elimination (Multiply First)

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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

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…

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.

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)

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

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.

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)

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

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

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.

Slide 11  Which variable is easier to eliminate?
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 3x + y = 4
 4x + 4y = 6
 x
 y
 6
 4

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.

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

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

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

Slide 16  Solve using elimination.
 2x – 3y = 1
 x + 2y = 3
 (2, 1)
 (1, 2)
 (5, 3)
 (1, 1)