Welcome back, everyone. So in an earlier video, I mentioned that step 2 of the elimination method was the trickiest step. This is where you had to look at your equations and figure out what to multiply them by, 1 or both of them, by some number, positive or negative, so that your x's and y coefficients would cancel out. This is a little bit of a trial and error method that we saw here. I'm gonna give you a little bit more information and a more systematic way to figure out how to multiply your equations in this method. And it really just comes down to looking at the coefficients of each equation. So what I'm gonna do is actually just gonna walk you through the 4 different possible scenarios that you'll see. We'll walk through each one of these examples, and I'm not gonna fully solve each one of them, but we're just gonna do steps 2 and 3 so that you can do the rest later on. Let's go ahead and get started here. So it really just comes down to the coefficients. Let's take a look at the first situation here.
If your coefficients of x or y are equal with opposite signs, then you actually just have to do nothing. You don't have to do anything, and you can just go ahead and start the addition process. So, really, if you have something like these two equations over here, \(7x + 13y = 12\), \(-7x + 2y = 18\), you don't have to do anything because these x's will just cancel already. So you'll eliminate the x's. The sum will become \(15y = 30\). And then you'll solve for this, and you'll see that \(y = 2\). And then you can just solve the rest from here.
Let's take a look at the second one here. What if the coefficients of x or y are equal with the same sign? Well, what happens here is if you have something like \(5x+7y = 17\) and \(6x+7y = 12\), these things are the same, but they're the same sign as well. So if you add them, they won't cancel out. All you have to do in these situations is you have to multiply either equation by negative one. It really doesn't matter which one you do it to because either way, you'll get now equal and opposite signs.
What I'm gonna do is I'm just gonna multiply this top equation over here by negative one. So if I drop this down, what happens is basically all the coefficients will flip sign. \(-5x - 7y = -17\). Now, over here I've got \(6x + 7y = 12\). Now, if you take a look here, we've basically just gotten back to this situation over here where we have equal and opposite signs, so we're done. All we have to do is just add them. What you'll see is that the y's will cancel, the \(-5x + 6x\) becomes \(x\), \(-17 + 12\) becomes \(-5\). And then from here, you have one of your answers, and you can already just go ahead and plug it and solve the other one.
Now let's move to the third situation here. But you may actually see some situations in which the coefficients of x or y will be factors of each other. Basically, it just means that they're evenly divisible by each other. Let's take a look at the situation over here. We have \(12x - 5y = 24\), \(3x - 2y = 6\). Now if you look at these 2 coefficients, they are not evenly divisible, but the \(12\) and the \(3\) are. So what are you supposed to do in this situation? Well, you'll multiply the equation with the smaller coefficient by whatever is the quotient between these coefficients, which is \(12 \div 3 = 4\).
So all I have to do is I have to multiply this equation here by \(-4\) because then you're gonna get your coefficients to cancel. So what this becomes over here is \(-12x + 8y = -24\). Now what happens is this basically just turns into the other situations. We'll see that the x's will cancel, leaving you only with the y's. Here we have \(8y - 5y = 3y\). And now over here, we have the \(24\) and \(-24\) that will cancel. So this just becomes \(0\), \(y = 0\), and then you can just finish the rest of the problem from there.
Now if any of these three situations don't apply, so in other words, if your coefficients of x and y are anything else, then one surefire method that will always work is you can always just multiply each equation by the other's coefficients. And you may exactly and you may have to put a plus or minus sign in there. So for example, let's say we have something like \(6x + 2y = 10\), and then we have \(-4x + 3y = 15\). So we can see that none of these things are evenly divisible into each other or factors. What we can do here is we can just multiply this equation over here to cancel out the x's. I'm gonna multiply this equation by \(4\), and I'm gonna multiply this equation over here by \(6\).
So what we're gonna see here is that now the first equation becomes \(24x + 8y = 40\). And then this over here will be \(-24x - 18y = -90\). Now what you'll see is that basically just turns into the other situations; the x's will cancel, leaving you with just the y's. And what we'll see here is you have \(8 - 18y = -10y\), and then \(40 + 90 = 50\). What you're gonna see here is that \(y = -5\), and then you could solve the rest of the problem from there. So really, this situation will actually always work, but it's always just a good idea to look for any of these other three apply. It's also just, you know, sometimes you may be able to look at the y's instead of the x's. That may be easier. So there's a bunch of different ways to do this. So hopefully this makes sense, folks. Let me know if you have any questions, and I'll see you in the next video.