Look carefully at the equations we have here. Firstly, they are very similar. Both equations have a single $x$ with the same signs(positive) and both equations have a single $y$ but with opposite signs. We can use either of these facts to solve this set of equations.

I am going to show how both approaches with give the right answer, but that the first is almost always the best one to use. Hopefully you will be able to see why as we proceed.

Gaussian elimination

Method 1

1. Write down the equations, one below the other, label them $(1)$ and $(2)$ and draw a line under them.
2. Subtract each part of the equations. $x-x=0$, $y --y=2y$ and $20-10=10$. The $y--y$ section is the reason why this method is not so great. Subtracting a negative is the same as adding a positive. However, this is very easy to make mistakes with.
3. We now know one of the values ($y=5$) and we will use substitution to find the other value. We will substitute the value into equation $(1)$, though we could use either of them. The reason I use equation $(1)$ is that it has fewer minus signs and minus signs are the biggest cause of errors in mathematics.

Method 2

1. Write down the equations as before.
2. Notice that if we add $+y$ to $-y$ we get $0$. And adding is much easier than subtracting, particularly subtracting negatives.
3. Other than that, it works the same way as before.

Note: It is almost always better to add than subtract, if possible.

Substitution method

1. The 2^nd equation $y=2x$, gives one variable in terms of the other. This means that we can substitute this into the 1^st equation, leaving us with a single variable ($x$).
2. Solve the equation for $x$.
3. Substitute into equation (2) to get the value for $y$.

1. Substitute equation $(2)$ into equation $(1)$. This is to replace the $y$ in the first equation by the expression $x-2$.
2. Note that in this situation, we must use brackets when we do the substitution.
3. Expand the brackets and solve for $x$.
4. Finally, substitute again into equation $(2)$ to find the value of $y$.

You should notice immediately that this example presents us with a new difficulty: there are $3x$s and $2x$s in the two equations, respectively. Likewise there are $+2y$ in the first and $-y$ in the second. This means that in the current form, we cannot proceed.

Fortunately, we have a way out of this problem. Remember the Golden Rule of Equations? We can do almost anything to an equation, so long as we do the same thing to both sides. Now if we were to multiply the second equation by $2$, then our equations can be solved.

1. Write down the equations.
2. Multiply eqn $(2) \times 2$ to make eqn $(3)$
3. Add eqns $(1)$ and $(3)$
4. Solve for $x$
5. Substitute to find the value of $y$.

1. This time there is no simple way to turn a $4$ to a $5$ or a $3$ to a $2$.
2. This means that we must balance both of the equations.
3. We will balance the $y$s by multiplying eqn $(1)$ by $2$ and eqn $(2)$ by $3$.
4. Add equations $3$ and $4$.
5. Substitute $x=3$ into eqn $(1)$ and solve to find the value of $y$
6. The labeling of the equations and making clear whether you are adding or subtracting, is essential, if you are to avoid making endless errors with this technique. You have been warned!