Welcome back, everyone. So we saw in an earlier video how to solve a system of equations by graphing. So we would graph these two lines, for example, and we would just find the place where they intersect, and that would give us our x and y values. But some problems, like the one we're gonna work out down here, will ask us to solve a system of equations without graphing. And they'll ask you specifically, in some cases, to solve it by using something called the substitution method. So that's what I want to show you how to do in this video. I want to show you the substitution method. I'm gonna show you it's actually a set of really straightforward steps here, these 5 steps. And we're gonna see a lot of stuff that we've already seen how to do before, like plugging in numbers for variables and expressions and things like that, and also just solving some simple equations for x and y. Let's go ahead and get started.

Now, first, I actually want to talk about why we even need this method in the first place. Remember, the solution to a system of equations is really just a set of numbers. So for these two equations, there's just going to be a two numbers, x and y equals something. And when I plug them into both of these equations, I'll get true statements. So if I can't graph these equations, can I just sit here guessing a bunch of numbers for x and y? And you totally could. The problem is it's gonna take you a long time because you might find a pair that works for one of the equations or the other, but not both of them. And so this kind of guess and check method isn't gonna be useful. I'm gonna show you a much more straightforward and systematic way to solve this.

The basic idea is that to solve these types of problems without graphing, we're just going to substitute one equation into another, and that's gonna make our equations simpler. That's why we call it the substitution method. Let's go ahead and get started. The first thing you're gonna do here is you're gonna choose the easiest equation to isolate x or y, and you're gonna call that equation a. So for example, this equation over here, y = 7x - 14 , is already isolated for y. So I'm just gonna call that my equation a. And by default, this is gonna be equation b. That's the first step.

The second thing you want to do is actually just solve that equation for x or y, whichever variable is the easiest to do it. It actually doesn't matter. You can solve for x or y. It really won't matter.

Now let's take a look at the 3rd step, which is the actual substitution. We're going to take this equation, a, and we're going to substitute it into b. What does that mean? We have an expression in this equation. We see that y is equal to 7x - 14. So wherever I see y in equation b, I'm actually just now going to replace it with that expression over here. So now what does this become? Well, equation b just becomes 2x - y. But now instead of y, I'm actually just going to replace it with the expression that I have over here, 7x - 14. And now this is gonna equal 4 over here.

Now all I have to do here is I just have to solve b. Notice how after the substitution, we've gone from 2 equations in which both of them have 2 variables that are unknown. And now in this equation, we see that we've eliminated we've we've sort of gotten rid of the y because we've substituted it. And now we only have x's, and we know exactly how to solve for that. So this really just becomes this is gonna become 2x - 7x + 14 = 4. Now we can solve for this. This really just becomes -5x, like this. And then over here, if we subtract 14 from both sides, what we're gonna see is that this is -5x = -10. And if we solve for this equation, we're just gonna get that x is equal to 2. So notice how now we've actually gotten one of our answers. We've gotten x equals 2. This is one of our numbers that if we plug it into one of the equations or either one of the equations, we'll get true answers or true statements.

So now that we've already gotten sort of the first half of our solution, which is x equals 2, how do we get the y value? Well, now that we're done with step number 3, we're gonna move on to step number 4. We're gonna plug the value that we just got, this x equals 2, back into either equation. It actually doesn't matter whether you plug it back into a or b, and then you just have to solve. So, for example, we're gonna take this x equals 2, and I'm just gonna go ahead and plug it back into equation a because it's already solved for y. So what you'll see here is if you plug this back into a, this says that y equals 7, and then instead of x, I'm actually just gonna write a 2 because that's what x equals. Right? It just equals 2 minus 14. So if you look at this, this actually just equals 14 minus 14, and what you'll see here is that y is equal to 0.

So here is the other half of my solution, x equals 2 and y equals 0. So these are the two numbers that if I plug them into these two equations, I'll get true statements for both. And if you don't believe me, we can actually move back to step number 5, which just says that you should check your answer by plugging the values into both equations. So let's go ahead and do that, step number 5. So for the blue equation, this says that y = 7x - 14, but now I'm actually just gonna replace the values with x and y. So instead of y, I plug in 0, and instead of x, I plug in 2. So does 0 equal 7 \times 2 - 14? And actually, we'll see that this does end up being a true statement because 0 = 14 - 14, and so in other words, 0 = 0. That's a true statement. Now that was equation A over here. Let's do the same exact thing for B. So for B, what we have is that 2x - y, except wherever I see x I'm going to plug 2 into - y, which is 0, does that equal 4? And we'll see that 4 minus 0 does actually equal 4, which is a true statement. So if you ever are just uncertain about the values you've gotten, you can always plug them back into your equation just to double check. So, anyway, that's the substitution method. Thanks for watching. Let me know if you have any questions.