Find the solution set of the two equations represented by 𝐿 one and 𝐿 two.
So 𝐿 one and 𝐿 two are lines. And when finding a solution, finding a solution from a graph is actually very straightforward. It’s something where the lines intersect. And they intersect at this point and that point is two, three. Therefore, two, three would be our solution set.
Now, to be safe, we could solve for the solution another way. If we could find the equation of 𝐿 one and the equation of 𝐿 two, we could use the equations to solve for 𝑥 and 𝑦. The equation of a line is 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope — the rise over the run — to the change in 𝑦 divided by the change in 𝑥 and 𝑏 is the 𝑦-intercept, where we cross the 𝑦-axis.
So let’s find the equation of 𝐿 one. Let’s first solve for 𝑏. Where does it cross the 𝑦-axis? It crosses it here at 𝑦 equals zero. And now for the slope, the rise over the run, so how much do we rise? We rise three spaces and we run two spaces. So we get 𝑦 equals three halves 𝑥 plus zero. So we don’t have to put the zero. We can just have 𝑦 equals three-halves 𝑥.
Now, for the other line — 𝐿 two — again let’s begin with 𝑏. We cross the 𝑦-axis at 𝑦 equals three. And now, how much do we rise? We rise nothing. We rise zero. And then to reach our destination of the other point, we ran two. But it really doesn’t matter how much we run because zero divided by anything is simply zero and zero times 𝑥 is zero and zero plus three is three. So the equation of that line is 𝑦 equals three.
So right there, we know that 𝑦 is equal to three, which we can see in our final answer. So if we wanted to solve for 𝑥, we could take three and plug it into the equation for 𝐿 one and then solve for 𝑥. And we should get two.
So to solve for 𝑥, let’s go head and multiply both sides of the equation by two. And two times three is six. So we have six equals three 𝑥. So now, we divide both sides of the equation by three and find that two is equal to 𝑥, just as we found before.
So once again, the solution set of the two equations represented by 𝐿 one and 𝐿 two is two, three.