2 Equations with no solution

On occasions we may encounter a pair of simultaneous equations which have no solution. Consider the following example.

Example 30

Show that the following pair of simultaneous equations have no solution.

10 x − 2 y = − 3 (7)

− 5 x + y = 1 (8)

Solution

Leaving Equation (7) unaltered and multiplying Equation (8) by 2 we find

10 x − 2 y = − 3
− 10 x + 2 y = 2

Adding these equations to eliminate x we find that y is eliminated as well:

10 x − 2 y = − 3
add − 10 x + 2 y = 2
0 x + 0 y = − 1

The last line ‘ 0 = − 1 ’ is clearly nonsense.

We say that Equations (7) and (8) are inconsistent and they have no solution .