yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Solving system with elimination | Algebra | Khan Academy


4m read
·Nov 11, 2024

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.

More Articles

View All
Pattern when dividing by tenths and hundredths
Let’s see if we can figure out what 2 divided by 0.1, or 1⁄10, is. Pause this video and see if you can figure that out. All right, now let’s work through it together. There are a couple of ways that we can approach it. One way is to think about everythin…
Official 2016 Trailer | Explorer
[Music] We’re heading out to the front, just a stone’s throw from Isis. This is the most at-risk coastline in the US. We could all go extinct. Back off! Don’t touch anything! Cult is not a negative thing if it c’s my head, and it does. These are criminals…
Going Through Hell (and Outsmarting the Devil)
What is hell? Is it a physical place that we go to in order to receive punishment? Or is it, perhaps, a human experience that our ancestors have tried to explain by using metaphors of terrifying worlds of torture? Many people these days would argue agains…
Writing arithmetic series in sigma notation
What I want to do in this video is get some practice writing Series in Sigma notation, and I have a series in front of us right over here. We have seven plus nine plus eleven, and we keep on adding all the way up to four hundred five. So first, let’s jus…
Consider THIS when navigating life's challenges
I hear so often that people say life can be difficult as though that’s an excuse for not being dealing with it well. Okay, so it’s difficult. So how do you approach it? Calm yourself down and then say, how does it work and how do I, uh, deal with it? One…
What kind of levers does the Speaker have? | US Government and Civics | Khan Academy
What kind of levers does the Speaker have in relation to the other House representatives? The Speaker has all kinds of levers, both formal and informal. In fact, a lot of them are informal. The Speaker can name a lot of people to the committees, particul…