Solving system with elimination | Algebra | Khan Academy

So we have a system of two linear equations here. This first equation, (x - 4y = 8), and the second equation, (-x + 3y = 11). Now what we're going to do is find an (x) and (y) pair that satisfies both of these equations. That's what solving the system actually means. As you might already have seen, there's a bunch of (x) and (y) pairs that satisfy this first equation. In fact, if you were to graph them, they would form a line.

There are a bunch of other (x) and (y) pairs that satisfy this other equation, the second equation, and if you were to graph them, it would form another line. So, if you find the (x) and (y) pair that satisfies both, that would be the intersection of the lines. So, let's do that.

Actually, I'm just going to rewrite the first equation over here, so I'm going to write (x - 4y = -18). We've already seen in algebra that as long as we do the same thing to both sides of the equation, we can maintain our equality. So, what if we were to add, and our goal here is to eliminate one of the variables. Therefore, we have one equation with one unknown.

So, what if we were to add (x + 3y) to the left-hand side here? So (x + 3y) well, that looks pretty good because an (x) and a (-x) are going to cancel out. We are going to be left with (-4y + 3y); well, that's just going to be (y).

So, by adding the left-hand side of this bottom equation to the left-hand side of the top equation, we were able to cancel out the (x)s. We had (x) and we had a (-x); that was very nice for us. So, what do we do on the right-hand side? We've already said that we have to add the same thing to both sides of an equation.

We might be tempted to just say, well, if I have to add the same thing to both sides, well, maybe I have to add (x + 3y) to that side. But that's not going to help us much. We're going to have (8 - (x + 3y)), and we would have introduced an (x) on the right-hand side of the equation.

But what if we could add something that's equivalent to (x + 3y) that does not introduce the (x) variable? Well, we know that the number (11) is equivalent to (x + 3y). How do we know that? Well, that second equation tells us that.

So once again, all I'm doing is I'm adding the same thing to both sides of that top equation. On the left, I'm expressing it as (x + 3y), but the second equation tells us that that (x + 3y) is going to be equal to (11). It's introducing that second constraint, and so let's add (11) to the right-hand side, which is, once again, I know I keep repeating, it's the same thing as (x + 3y).

So (8 + 11) is (19), and since we added the same thing to both sides, the equality still holds. We get (-y = -19) or divide both sides by (-1) or multiply both sides by (-1). So, multiply both sides by (-1), we get (y = 19).

So we have the (y) coordinate of the (xy) pair that satisfies both of these. Now how do we find the (x)? Well, we can just substitute this (y = 19) into either one of these. When (y = 19), we should get the same (x) regardless of which equation we use.

So let's use the top equation. We know that (x - 4 \times 19 = 8). So, (4 \times 19 = 76). So let's see. To solve for (x), I could add (76) to both sides.

So, add (76) to both sides on the left-hand side; (−76) and (+76) cancel out. I'm just left with (x), and on the right-hand side, I get (8 + 76) which is (84). So there you have it. I have the (xy) pairs or the (xy) pair that satisfies both: (x = 84) and (y = 19).

I could write it here as coordinates: ((84, 19)). Notice what I just did here. I encourage you to substitute (y = 19) here, and you'll also get (x) as (84). Either way, you would have come to (x = 84).

And to visualize what is going on here, let's visualize it really fast. Let me draw some coordinate axes. Whoops, I meant to draw a straighter line than that. All right, there you go. So let's say that is our (y) axis and that is our (x) axis.

Then let's see. The top line is going to look something like this. It's going to look something like this. And then that bottom equation is going to look something like that.

Let me draw a little bit nicer than that. It's going to look something like this, something like that. Let me draw that bottom one here so you see the point of intersection.

The point of intersection right over here is an (xy) pair that satisfies both of these equations, and that we just saw happens when (x = 84) and (y = 19). Once again, this white line is all the (xy) pairs that satisfy the top equation. This orange line represents all the (xy) pairs that satisfy the orange equation.

Where they intersect, that point is on both lines and satisfies both equations. Once again, take (x = 84), (y = 19); substitute it back into either one of these, and you will see that it holds.