## Using Matrices to Solve Simultaneous Equations

If you already know how to solve simultaneous equations then you may well wonder why people use matrices to solve them. The fact is, while simple equations with two unknowns x and y are quite easy to solve, as the number of unknowns increases so does the number of equations we have to solve. The number of calculations we have to do though, increases by far more than the number of unknowns we have to find. But the matrix method remains generally the same , and is suitable for computers to crunch on.

The method goes like this. We write the equations we have to solve in matrix form: where is a matrix, is the column vector and b is the column vector Then we find the inverse of the matrix and multiply on the left by  (we have used ).

Example: Solve the simultaneous equations

4x+3y=9

7x+6y=10

First we write the problem in matrix form: Then multiply on the left by The inverse of a 2 by 2 matrix is given by  Example: Solve the simultaneous equations

4x+3y=5

3x+4y=8

First we write the problem in matrix form: Then multiply on the left by   