How to solve a 2x2 system of linear equations?

For example, solve the following 2x2 system of linear equations:

Substitution Method

Express one variable in terms of the other from the second equation:

  • x - 2y = -4 → x = 2y - 4

Substitute x = 2y - 4 into the first equation:

  • 2(2y - 4) + 3y = 13
  • 4y - 8 + 3y = 13
  • 7y - 8 = 13 → 7y = 21 → y = 3

Substitute y = 3 into the expression x = 2y - 4:

  • x = 2(3) - 4 = 6 - 4 = 2

The solution to the system of equations is: (x, y) = (2, 3)

Algebraic Addition Method

Multiply coefficients to eliminate one variable. Multiply the second equation by 2:

The system of equations becomes:
2x + 3y = 13 and 2x - 4y = -8

Subtract the two equations to eliminate x:

  • (2x + 3y) - (2x - 4y) = 13 - (-8)
  • 7y = 21 → y = 3

Substitute y = 3 into the equation x - 2y = -4:

  • x - 2(3) = -4
  • x - 6 = -4 → x = 2

The solution to the system of equations is: (x, y) = (2, 3)