In conjunction to using simultaneous equations in the form of matrices, the inverse matrix is used in order to find a relationship between the two before solving the pair of simultaneous equations. To see how, it is best to go through an example.

### Application

In solving the equations

$2x + 4y = 4$
$7x + 3y = 9$

The follow set of matrices are distinguished:

The matrix product that represents the simultaneous equations is:

$AX = C$

If the inverse of $A, A^{-1}$ the following relationship is determined:

$A^{-1}AX=A^{-1}C$
$IX = A^{-1}C$
$X = A^{-1}C$

Therefore, In order to solve the simultaneous equations for matrix X, the inverse of A must be found and pre-multiplied to matrix C. Note: It is A^(-1)*C not C*A^(-1). The inverse of A is multiplied BEFORE C.

The solution to the pair of simultaneous equations is therefore
$x = \dfrac{63}{59}$,  $y = \dfrac{10}{59}$

## No unique solutions

However, sometimes simultaneous equations do not have solutions. This occurs when the determinant = 0. There are two reasons behind this:

### Inconsistent equations

It should be reminded that the solution to a pair of simultaneous equations represent the coordinates that the two lines cross. In the case of inconsistent equations, the two lines used in the simultaneous equations are parallel and therefore will never intercept with each other.

For example:

$2x + 4y = 4$
$4x + 8y = 9$

Both of these equations have a gradient of -0.5 but have different y-intercepts. This means the lines will never meet with each other.

### Dependent equations

Rather than having no solution, sometimes there are indefinitely many solutions and hence no unique solution. This occurs when the two simultaneous equations lie on the same line.

For example:

$2x + 4y = 4$
$4x + 8y = 8$

When plotted, these two lines are exactly on top of each other. This means that the equations are dependent and that there is no unique solution.