How to Use Matrices to Solve Simultaneous Equations

A TI-83 graphing calculator, useful for matrix calculations

Matrices are potentially intimidating mathematical concepts to learn, but they have countless uses. Matrices are essentially just grid-based representations of numeric data and can be used to organize and solve problems that might otherwise take much longer. They are especially useful in solving a system of linear equations.

Things You'll Need

• Calculator

Instructions

1. Rewrite your equations so that all similar variables are aligned. For example, for the equations x + 2y = 5 and 5y = 26 - 3x, write the following;

   x + 2y = 5
   3x + 5y = 26

2. Write a matrix using the coefficients of each equation as the rows of the matrix. The coefficients are the numbers by which the variables are multiplied. For example, for
   x + 2y = 5 and 3x + 5y = 26, write the following:

   [ 1 2
   3 5 ]

3. Enter this matrix into your calculator. For example, on a TI-83, press the "matrix" button and enter the relevant numbers into the desired matrix.

4. Construct a similar matrix for the answers to your equations. For example, for x + 2y = 5 and 3x + 5y = 26, write the following:

   [ 5
   26 ]

   Enter this matrix into another slot on your calculator.

5. Multiply the inverse of your first matrix by the second. Select the first matrix and raise it to the -1 power, and multiply that by the second. The resulting matrix consists of the solutions to your equations. For example, for x + 2y = 5 and 3x + 5y = 26

   [ 1 2 ^ -1 * [ 5 = [ 27
   3 5 ] 26 ] -11 ]

   The solutions to the equations are x = 27 and y = -11.

References

• Algebra Lab; Solving Systems of Equations with Matrices