If you solve a system of equations such as "find x and y" using algebra alone, it can be hard to see what the answer actually means. In fact, the answer is the coordinates of the single point where the two linear-function graphs meet. Move the two lines with the sliders, watch the intersection move, and check for yourself whether its coordinates are the same as the solution you get by solving the equations.
Why is the intersection the solution to the equations? Every point on line 1 satisfies the first equation, and every point on line 2 satisfies the second equation. The point where the two lines meet is the only (x, y) that satisfies both equations at the same time, so that point is the solution to the system of equations.
Try making the slopes (a₁, a₂) the same — the two lines become parallel and never meet. In this case, the system of equations has no solution. On the other hand, if you make the intercepts (b₁, b₂) the same as well, the two lines completely overlap, giving infinitely many solutions.