# Solving a Linear System in Two Variables

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:

- Write each equation in the same format, with variables in the same order in each, and all set equal to the constant.

Subtracting 3*y*from each side of the first equation puts the system into “*x, y*equals constant” order.

- 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.

- 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.

- 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).