What is the best way to solve simultaneous equations?

The easiest way to solve simultaneous equations is to label the equations so you clearly follow them throughout.

For example, say you are asked to solve the unknowns for the simultaneous equations 3x-y=8 and 2x+y=7 

Label the first equation 1.) and the second 2.), like so:

1.) 3x-y=8 

2.) 2x+y=7

You now need to perform an operation to eliminate one of the unknowns from the equations. It is always easiest to elimate the unknown which has the same multiple in each equation.

So here, we would work to elimate y. Looking at the signs we can see that equation 1.) has a  negative y and 2.) a positive, so all we have to do to eliminate y is the add the equation 2.) to equation 1.), like so:

1.) + 2.) : 5x=15

Now, all you have to do is solve this like a regular equation, so by dividing 15 by 5 we get x=3.

The next step is substitute the known x value into either one of the original equations. I will use both here to show that either works:

1.) 3(3)-y=8

so 9-y=8, and hence y=1. 

2.) 2(3)+y=7

so 6+y=7, and again, y=1

It is always useful to substitute your first known variable into one equation, and then substitute both variables into the second, as a check to see if you are correct.