This section reviews situations where graphing, substitution, and elimination will not find a point of intersection when solving the system of equations.

There are two special cases when solving linear systems of equations.

The first case occurs when solving the systems algebraically. The variables are eliminated, and the left side of the equation does not equal the right side of the equation. Then there is no solution, the lines are parallel.

Example: Solve

-2x + 2y = 6
-x + y = -5

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

• Multiply the second equation by -2 to create additive inverses:________________. Interactive popup. Assistance may be required.

  Check Your Answer

  -2(-x + y) = -5(-2)
  2x – 2y = 10
  
• Add the new equation with the first equation:_______________. Interactive popup. Assistance may be required.

  Check Your Answer

  -2x + 2y = 6

  2x – 2y = 10

  ---

  0 ≠ 16
  
• Notice all the variables are inverses of each other resulting in the left side of the equation equaling __________ while the right side's sum is __________. Interactive popup. Assistance may be required.

  Check Your Answer

  0 while the right side's sum is 16
• The system has _________ solution and the lines are _________. Interactive popup. Assistance may be required.

  Check Your Answer

  no solution and the lines are parallel

The second case occurs when solving the systems algebraically. The variables and the constants are eliminated, and both sides of the equation equal zero. Then any number can be a solution; the lines are coinciding.

Example: Solve

3x – 4y = 12
-6x + 8y = -24

• Multiply the first equation by 2 to create additive inverses:_________________. Interactive popup. Assistance may be required.

  Check Your Answer

  2(3x – 4y) = 12(2) ⇒ 6x – 8y = 24
• Add the new equation to the second equation:________________. Interactive popup. Assistance may be required.

  Check Your Answer

  6x – 8y = 24

  -6x + 8y = -24

  ---

  0 = 0
  
• Notice all the variables are inverses of each other, and the left and right sides of the equation both equal ____________. Interactive popup. Assistance may be required.

  Check Your Answer

  zero
• The system has _____________ solutions and the lines are ________________. Interactive popup. Assistance may be required.

  Check Your Answer

  infinitely many solutions and the lines are coinciding