 So, we've spent quite a bit of time learning about how to solve single equations. What about taking a look at systems of equations? What's a system of equations, you ask? A system of equations consists of two or more equations. A solution to a system of equations consists of values of the variables that make all of the equations true. For example, does x equals 3, y equals 5 solve the system of equations to x plus 5, y equals 31, y minus 3, x equals 43? So, we'll check to see if x equals 3, y equals 5 makes all of the equations true. Checking the first equation equals means replace, so we'll replace x with 3 and y with 5 and see if we get a true statement. And we do! So, x equals 3, y equals 5 solves the equation to x plus 5, y equals 31. But in order to solve the system of equations, it has to solve both equations. So, we'll substitute x equals 3, y equals 5 into the second equation. And check to see if the resulting statement is true. And it's true! I mean, false! And since this is a false statement, x equals 3, y equals 5 does not solve the second equation, so it is not a solution to the system of equations. Again, it's useful to work back and forth between algebra and geometry if our equations are in two variables that correspond to a graph. So, what would it mean that we have a solution to a system of equations? Well, suppose x, y solves two equations. Since x, y solves the first equation, it is a point on the graph of the first equation. And since x, y solves the second equation, it is a point on the graph of the second equation. So, x, y is a point on the graph of both equations. This means that the algebraic solution to a system of equations corresponds to the geometric intersection of two graphs. For example, suppose I have the graph of two equations. How do we know these have the graphs of these two equations? Because they've been labeled. They've been labeled. So remember, if I have a point on the graph of 3x minus 5y equals 13, it automatically satisfies the equation 3x minus 5y equals 13. And likewise, any point on the graph of 7x plus 3y equals 1 satisfies that equation. If I want a point that satisfies both equations, I need a point that's on both lines, and that means I have to find the intersection point. So, we see that the graphs intersect at the point 1, negative 2. So, x equals 1, y equals negative 2 is a solution to the system of equations. And we can solve any system of equations this way. So, if I want to solve the system of equations, I can try graphing both equations, then find the coordinates of the intersection point, which are very clearly something more than 1, but less than 2, and something a little less than 0. And this is the problem with trying to solve equations by graphing. We have to be able to read the coordinates of an intersection point with unlimited accuracy. But in general, we can't do that. So, what we really want is an algebraic method of finding these intersection points, an algebraic method of solving systems of equations. We'll take a look at that next.