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Solving systems of linear equations — Basic example

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- [Instructor] Y is equal to 3x, x is equal to 3y. Consider the system of equations above. How many solutions does this system have? Well, when I think about solution systems, I always imagine each of the equations in our system as kind of describing a line. And if the two lines intersect, you're going to have one solution, and if they don't intersect, you're going to have no solutions, and if they end up being the same line, so if they end up being the same line, you have an infinite number of solutions. So let's see, we can immediately rule out, 'cause these are linear, these are linear equations, two solutions. The only way you're gonna have two solutions is if you have, if one of these lines, at least one of them curves. If one of them did something like this, then you could have two solutions, right there, and right there, but these are both linear equations. So we could rule out having two solutions. But let's just think about these up here. Well what I like to do, is I like to, let's solve them both explicitly for Y. That's at least, how I like to visualize lines. This first one up here is already written explicitly in terms of Y. The second one over here, let's divide both sides by three. And we get Y is equal to X over three. So this first one is y is equal to 3x, the second one is y is equal to x over three, or we could say 1/3, 1/3 X. These lines, they don't have the same slope, they don't have the same slope, so they're not going to be parallel, and they're not going to be the same line. So these are going to intersect. These are going to have one solution, exactly one solution. If you wanted to visualize this on the coordinate axis, I'll od it really fast, if that's our Y axis. This is our X axis, X Y. This first line, right over here, this is going to have a slope of three. So it's gonna be a pretty steep line. It's gonna look something like this. And then the second line right over here is going to have a slope of 1/3. So it's going to look something, it's gonna look something like this. These two clearly intersect. And actually, now that I look at it, one way to realize that you're gonna have at least one solution, and then you'll see it, you can see it where they're intersecting right over here, the solution is when X and Y both are equal to zero. But since you wanna rule out that hey, maybe it has an infinitely many solutions, is look, just think about what these lines represent. This first one, very steep line right over here. The second one, a much less steep line. But they're different lines, so they're not the same line, and they don't have the same slope, so they're not going to be parallel. So they're going to intersect in exactly one point.