Graphical method
In the figure below, I have graphed the system of equations,
$\begin{align*} 2x-y&=3\\ 3x+2y&=8 \end{align*}$
The two graphs intersect at the point $(2,1)$, which is the solution of the system. So, we can find the solution of a system of equations from a graph: the solution is the point where the two graphs intersect.
Note that not every system of equations has a solution. Some do, some don't, and some have infinitely many solutions. If the graphs of a system's equations are parallel — that is, they do not intersect — then the system has no solution. If the graphs overlap, then the system has infinitely many solutions. In this case, the equations are dependent; that is, the two equations are the same but written in different forms.
Example 1: Solve the following system of equations,
$\begin{align*} -x-2y&=-4\\ x+2y&=2 \end{align*}$
The graphs for these two equations are in the figure below.
In the figure, the graphs are parallel, so they do not intersect; therefore, the system of equations has no solution.
Example 2: Solve the system of equations,
$\begin{align*} -2y=x-4\\ x+2y=4 \end{align*}$
The graphs of the above equations are in the figure below.
Here, the graphs of the two equations overlap, so there are infinitely many solutions. Overlapping graphs mean that both equations are the same, just written in different forms.