Common Core Standards: Math
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
By now, students should know that a linear equation given by y = mx + b (or a variation of it) and makes a line on a graph. They should also know that a system of linear equations means that we have more than one y = mx + b equation in the mix. So far, so good.
Solving a system of linear equations means finding the point at which the two (or more?) lines intersect. This happens when the same set of x and y values satisfy all the linear equations in the system.
What about lines that never intersect? Well, they're parallel, for starters. But that just means that the system of parallel lines will have no solution. (It may be helpful to tell your students that "No solution" is a legitimate answer, but only after they've done the work to prove it. Otherwise you'll get "No solution" as the answer to every homework problem.)
Students should be shown that a system of linear equations can be solved either through graphs or straight algebra, but that these two methods arrive at the same answer because they mean the same thing. The goal is to find the point at which the two lines intersect.
Graphically, solving a system of two linear equations is easy. Using the slopes and y-intercepts of both lines, students can graph each line individually and find the intersection point. It's pretty and simple (and pretty simple), but not always accurate visually. For instance, what's the solution to the following system of equations?
Is it at point (6.2, 17)? Or maybe (6.3, 17.5)? It's difficult to tell exactly. Very helpful visually, but a bit less useful in terms of coming up with numerical answers.
It'd be helpful to know that the two lines above have equations of y = 2x + 5 and y = 3x – 1.25. With the linear equations, students should be able to graph the lines on a coordinate plane as well as solve them exactly using substitution. When the lines intersect, the x and y values in both linear equations are the same. That means we can set the two equations equal to each other.
For instance, we can say that 3x – 1.25 = 2x + 5 because both the y coordinates have to be equal. (By the way, that gives x = 6.25 as our answer. If we substitute 6.25 for x back into either equation, we should get 17.5.)
If students are struggling with these concepts, emphasizing the connection between the visual graphs and the algebraic calculations should help cement their understanding.