Problem solving $2\times 2$ equation system $$\left\{\begin{align}
2x + 3y &= 10 \\
4x -  y &= - 1
\end{align}\right.
$$
1) I don't get why I have to multiply first equation by $-2$ and not $+2$
2) How does $x=\frac{1}{2}$
I'm only $15$ so try and explain as simple as possible. Thanks.

it don't matter whether you multiply by the first equation by $2$ or $-2.$ what you want to do is to go from a set of simultaneous equations of two variables $x, y$ to an equation with just one variable. in this example you are trying to eliminate $x$ so that you end up with an equation in only $y.$
if we multiply the first equation by $2,$  we have 
$$\begin{align} 4x + 6y &= 20\\4x-y &=-1. \end{align}$$
subtracting $(2)$  from $(1),$  we have $$7y = 21 $$ which give you $y = 3.$  putting this in one of the original equations 
$$4x-y = -1 \to 4x -3 = -1 \to 4x = -1+3 = 2 .$$ that gives you $x = 1/2.$
had you multiplied by $(-2),$  you would have ended up with $$\begin{align} -4x - 6y &= -20\\4x-y &=-1. \end{align}$$
now instead of subtracting, you will add the two and get $y = 3$ again.
