If you have a way to graph ax+by=c electronically, here is a fun way to teach solving systems of equations by linear combination.

Say you want to solve

Graph the two equations:

The solution is immediately obvious at the intersection: x = 6, y = -2. But we’re just getting started.

Notice that if you graph a multiple of one of the equations, you get the same graph as the original. (For example, 4x+6y=12 will have the same graph as 2x+3y=6.) This deserves a brief discussion.

But here is where it gets interesting: *if you graph a multiple of the first equation, plus a multiple of the second equation, the resulting line still passes through the original intersection.* For example, by adding the two original equations, I get 3x + y = 16. Here is the graph with the additional line:

Do this many times, to make sure everyone understands what’s going on. For example, subtract the equations, or add twice the first equation to three times the second, and so on. These are called linear combinations of the two equations. Of course, you may discuss why the new line always passes through the intersection.

Now start over with a new system, and challenge the students to come up with a linear combination that will create a vertical line through their intersection, and one that will create a horizontal line through it. (One teacher I knew years ago called this activity “add till it’s plaid”.) Graph all suggestions, until someone figures it out. Give hints if necessary.Once someone has succeeded, discuss how that feat was accomplished.

Finally do this with yet another system, *but no graphing*. Tada! The method that was used to get the horizontal and vertical line is one way to solve the system.

–Henri

PS: Chris Bolognese has created a Desmos Activity Builder lesson based on this blog post. Check it out!

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