 So there's two ways of solving systems of equations and one of them is called solving by substitution. So a system of equations consists of several equations. For example, the equations 3x plus 5 equals 11, 2x minus 1 equals 8 forms a system of one, two equations in one variable, x is in both. So that's a system of two equations in one variable. Or I might have a system that looks something like this, 3x minus y equals 8, 2x plus y equals 7. That's one, two equations in one, two variables, x and y. So I have a system of two equations in two variables. Now a solution is going to consist of values of the variables. All the variables that are involved, whether it's one or two variables or three or 25, it's going to make all of the equations simultaneously true. And that's an important thing to keep in mind. The equations not only have to be true individually, but for that set of x and y values, all equations have to be true. So for example, if x equals 2, 3x plus 5 does in fact equal 11. But 2x minus 1 is not equal to 8. So this equation is true. This equation is not. These two equations are not simultaneously true for x equals 2. So x equals 2 is not a solution to the system of equations. If x equals 3, y equals 1, then 3x minus y equals 8. That's true. 2x plus y, that's 6 plus 1 equals 7. That's also true. And so that means that 4x equals 3, y equals 1. Both of these equations are simultaneously true. So that means x equals 3, y equals 1 is a solution to the system of equations. 3x minus y equals 8, 2x plus y equals 7. So how do you solve systems of equations? Well, again, there's two primary methods. And in this case, we'll look at one, which is known as the substitution method. What we want to do is we want to express one variable expression in terms of the other variables. We want to substitute into an equation, and we'll continue substituting variables until we have an equation in just a single variable, and then we'll solve that equation. And we'll use our equations to solve for the remaining variables. So for example, let's take a look at the system of equations, y equals 3x minus 5, y equals x plus 9. So I want to solve for one variable in terms of the others. So, well, okay, did it. So here I've solved for y in terms of x, or I've solved for y in terms of x. I've given one variable in terms of the other, so I don't actually have to do much solving here. I do want to substitute into one of the other equations. So here I have the equation y equals x plus 9, and I'm going to substitute y for something that it's equal to. Now, I know that y is the same as x plus 9, but I don't want to replace this y with x plus 9, because then I have x plus 9 equals x plus 9, and that doesn't tell me anything new. But I also know that y is the same thing as 3x minus 5. So any place I see y, I can replace it with 3x minus 5. So, well, here's a y. I'm going to replace this y with 3x minus 5, and there's my substitution. And now I have an equation in one variable, and I have a whole bunch of methods of solving such equations. So I'll solve this equation, I'll solve this equation, I'll solve this equation, and I get solution x equal to 7. Now, remember that this is a system of one, two variables, x and y, and the solution is going to require that I specify both variables. So I know that x equals 7, so I do want to find the other variable, y, and so I know that y is the same thing as 3x minus 5. So if I want to know what y is, I want to find out what 3x minus 5 is going to be, and I'll substitute x equals 7 into that equation, and after all that settles, I find that yi is equal to 16, and so my solution x equals 7, y equals 16.