Sometimes it is necessary to multiply one or both equations so that the coefficients of one of the variables in each equation is the same, apart from possibly a sign change.

e.g. Solve the simultaneous equations

If we multiply equation (2) by 2 we will get the same coefficient of y in both equations so that we can eliminate y from the equations.
This gives

Subtracting equation (1) from equation (3) gives x = 5.

To find the value of y we substitute for x in equation (1). So 15 + 6y = 21

We can then solve this equation to get y = 1.

The solution to the original equations is therefore x = 5, and y = 1.

e.g. Solve the simultaneous equations

In this example we could multiply the first equation by 5 and the second one by 2.

WHY ?