# In Media Res, or a running start

Let’s look at a system of two equations in two unknowns. We have encountered this topic many times in earlier years, and we have a number of ways to ‘solve’ the system.

$2x - y = 0 \qquad \cdots$ (1)

$-x + 2y = 3 \qquad \cdots$ (2)

Before we go on, let’s solve this system of equations in two different ways that we already know about.

First Method. If we had seen these two equations in middle school, we would have said “Those are the equations of two lines. Two (non-parallel) lines intersect in a point.” And you would be right. We can indeed graph these two lines. One goes through the origin, while the other does not. Using the graphical method, we can identify the point of intersection as the point $(1,2)$. In other words, the point $(1,2)$ is the one and only point that solves both of the equations we started with.

Second Method. (You may know this as Gaussian Elimination; if not, don’t worry ). Note that by solving this system of equations, we are finding the intersection point of two lines, just like before.

If we multiply equation (2) by 2, we can then add equations (1) and (2) together in order to eliminate the $x$ term. We are left with

$3y = 6$ which we can solve by inspection as $y = 2$.

Knowing that $y = 2$, we can substitute for $y$ in either of the equations we started with:

$2x - 2 = 0$, or $x = 1$, giving us the full solution, or point of intersection as $(1,2)$.