Solving a Linear System in Two Variables

By Mary Jane Sterling

Most of the applications in finite math that involve mathematical statements are of the linear variety. You can use an augmented matrix to solve a system of linear equations in two variables. For example, say you start with these two equations:


To solve this system, you would follow these steps:

  1. Write each equation in the same format, with variables in the same order in each, and all set equal to the constant.
    Subtracting 3y from each side of the first equation puts the system into “x, y equals constant” order.
  2. Write an augmented matrix with the coefficients of the variables as elements, the coefficients of the same variables under one another, and the constants in a column to the right, separated by a vertical bar. Replace any missing variables in the equation with 0.
  3. Perform row operations until the matrix consists of an identity matrix on the left of the vertical bar.
    Create a 0 below the 1 in the upper-left corner.
    Multiply by the reciprocal of 11 to make the element in the second row, second column a 1.
    And, finally, create a 0 above the 1.
  4. Read the solution from the numbers in the vertical column on the right; each value corresponds to the position of the 1 in the matrix to the left.

The 1 in the first row corresponds to the x variable, and the value in the right column is 4, so this tells you that x = 4. The 1 in the second row corresponds to the variable y, and the number in the right column is –2, so y = –2. The answer, written as the coordinates of a point, is (4, –2).