7.2, 7.3 Solving by Substitution and Elimination

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7.2, 7.3 Solving by Substitution and Elimination
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Solving by Substitution

• 1) Use one equation to get either x or y by
  itself
• Substitute what we just found in step 1 to the
  other equation
• Solve for one unknown
• Substitute the answer from step 3 to either of
  the original equation to solve for the other
  unknown
• Check your answer

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• x y 4
• x y 2
• Step 1) x is already by itself x y 2
• Step 2) Substitute to the first equation
• x y 4
• (y 2) y 4
• Step 3) 2y 2 4
• 2y 4 2 2
• y 1
• Step 4) x y 2
• x 1 2 3
• Solution (3, 1)

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• 2x y 6
• 3x 4y 4
• Step 1) get y by itself 2x y 6 so y -2x
  6
• Step 2) Substitute to the second equation
• 3x 4y 4
• 3x 4(-2x 6) 4
• Step 3) 3x 8x 24 4
• -5x 24 4
• -5x -20
• x 4
• Step 4) 2x y 6
• 2(4)y 6 so 8 y 6, y -2
• Solution (4, -2)

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• More practice (Make sure you check your answers)
• 1) 3x 4y 14
• 5x y 8
• Answer (2,-2)
• 2) 2x 3y 0
• -4x 3y -1
• Get x by itself 2x 3y 0 so x (3/2) y
• Substitute into -4x 3y -1
• -4(3/2)y 3y -1
• -6y 3y -1
• - 3y -1
  so y 1/3
• Solve for x Use the first equation 2x 3y
  0
• 
  2x 3(1/3) 0
• 
  2x 1 0
• 
  x ½
• Answer (1/2 , 1/3)

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Example 1

• The perimeter of a basketball court is 288 ft.
  The length is 40 ft longer than the width. Find
  the dimensions of the court.

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Example 2

• The sum of two numbers is 51. One number is 27
  more than the other. Find the numbers.

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Solve by Elimination

1. Multiply some numbers to either or both equations
   to get 2 opposite terms (For example 2x and
   -2x)
2. Add equations to eliminate one variable
3. Solve for 1 unknown
4. Substitute the answer from step 3 to either of
   the original equation to solve for the other
   unknown
5. Check your answer

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• Solve by elimination
• 2x 3y 0
• -4x 3y -1
• Ignore step 1 since we already have 2 opposite
  terms -3y and 3y
• Step 2 3 Add 2 equations to eliminate y and
  solve for x
• 2x 3y 0
• -4x 3y -1
• -2x -1
• x 1/2
• Step 4 Choose the first equation 2x 3y 0
• 
  2(1/2) 3y 0
• 
  1 - 3y 0
• 
  - 3y -1
• 
  y 1/3
• Answer ( 1/2, 1/3)

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• Ex2) 2x 2y 2
• 3x y 1
• Step 1 Multiply 2 to the second equation
• 2x 2y 2
• (2) 3x y 1
• 2x 2y 2
• 6x 2y 2
• 8x 4 (step 2 and 3)
• x 1/2
• Step 4 Choose 3x y 1
• 3(1/2) y 1
• 3/2 - y 1
• - y 1
  (3/2) - ½
• y ½
• Answer (1/2, 1/2)

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• Ex3) 2x 3y 8
• -3x 2y 1
• Step 1 Multiply 3 to the 1st equation and 2 to
  the 2nd equation
• (3) 2x 3y 8
• (2) -3x 2y 1
• 6x 9y 24
• -6x 4y 2
• 13y 26 (step 2 and 3)
• y 2
• Step 4 Choose -3x 2y 1
• -3x 2(2) 1
• -3x 4 1
• -3x 1 -4
  -3
• x 1
• Answer (1,2)

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Practice

• Solve by elimination
• 1) 3x 4y 14
• 5x y 8
• Answer (2,-2)
• 3x 2y 7
• 6x 4y 14
• Answer Infinite many solutions