In this section, solving systems of equations by elimination is explored.

When graphing or substitution cannot be easily done, elimination is used to solve a system of equations.

Example: Solve

8x + 11y = 37
2x – 11y = -7

Fill in the blanks below. Click "check your answer" to reveal the answers.

• Add the two equations:______________. Interactive popup. Assistance may be required.

  Check Your Answer

  8x + 11y = 37
  2x – 11y = -7
  

  ---

  10x = 30
  
• Solve for x:______________________. Interactive popup. Assistance may be required.

  Check Your Answer

  10x = 30 ⇒ x = 3
• To find the y-value, substitute in 3 for x into one of the equations: _____________. Interactive popup. Assistance may be required.

  Check Your Answer

  8(3) + 11y = 37
• Solve for y:_________________. Interactive popup. Assistance may be required.

  Check Your Answer

  24 + 11y = 37
  24 - 24 + 11y = 37 - 24
  11y = 13
  y = 13 11
  
• The solution to the system of equations is the ordered pair ____________. Interactive popup. Assistance may be required.

  Check Your Answer

  (3, 13 11 )

What if adding or subtracting does not eliminate a variable?

Example: Solve

3x – y = 8
x + 2y = 5

• If the equations were added, a variable would ___________ be eliminated. Interactive popup. Assistance may be required.

  Check Your Answer

  not
• If the y in the first equation were changed to______________, then the y variables would be additive inverses and can be ________________. Interactive popup. Assistance may be required.

  Check Your Answer

  If the y in the first equation were changed to -2y, then the y variables would be additive inverses and can be eliminated.
• Multiply the entire first equation by 2: _____________________. Interactive popup. Assistance may be required.

  Check Your Answer

  2(3x – y) = 8(2) ⇒ 6x – 2y = 16
• Rewrite the system of equations replacing the first equation with the new equation:_____________________. Interactive popup. Assistance may be required.

  Check Your Answer

  6x – 2y = 16
  x + 2y = 5
  
• Add the equations:____________________. Interactive popup. Assistance may be required.

  Check Your Answer

  6x – 2y = 16

  x + 2y = 5

  ---

  7x = 21

  
• Solve for x:_________________. Interactive popup. Assistance may be required.

  Check Your Answer

  7x = 21
  7 7 x = 21 7
  x = 3
  
• Find the y-value by substituting in 3 for x in either equation: Interactive popup. Assistance may be required.

  Check Your Answer

  x + 2y = 5
  3 + 2y = 5
  3 – 3 + 2y = 5 – 3
  2y = 2
  y = 1
  
• The solution to the system of equations is the ordered pair __________. Interactive popup. Assistance may be required.

  Check Your Answer

  (3, 1)

Decide if the following equations can be solved with simple elimination or if they must be transformed by multiplication first. Click on the blanks to reveal the answers.

1. 3x – 2y = 6
   6x + 2y = 8 Interactive button. Assistance may be required. __________ simple elimination
2. 4x – 6y = 12
   2x – 2y = 5 Interactive button. Assistance may be required. __________ multiplication and then eliminate
3. 5x + 7y = 14
   10x – 7y = -4 Interactive button. Assistance may be required. __________ simple elimination
4. 8x – 12y = 11
   -4x + -5y = 13 Interactive button. Assistance may be required. __________ multiplication and then eliminate

• If a system of equations is written in standard form, if possible Interactive button. Assistance may be required. __________ eliminate one of the variables by Interactive button. Assistance may be required. __________ adding the two equations together.
• If a variable cannot be Interactive button. Assistance may be required. __________ eliminated by adding the two equations together, use the multiplication property of equality to create an Interactive button. Assistance may be required. __________ additive inverse for one of the variables.
• Add the system of equations together. Now there is Interactive button. Assistance may be required. __________ one equation with Interactive button. Assistance may be required. __________ one variable.
• Solve for the remaining Interactive button. Assistance may be required. __________ variable, then Interactive button. Assistance may be required. __________ substitute this value back into one of the equations to find the second variable.