Determine the Number of Solutions in a System Lesson Plan

Number sense routine

Start your lesson by asking students which of the three graphs doesn’t belong. 

Have them think silently first, then share with the class. This will lead to awesome conversations, and possibly things we didn’t even think about!

No solution

Then begin to ask more targeted questions about each graph. The first example is below:

We want students to come up with the ways these two lines are similar and different. You can ask, “would any coordinate point satisfy both of these equations?” Ultimately, we’re having students figure out that since these lines will never touch, there are no solutions! 

One solution

Students have already learned how to graph systems with one solution, so this next one will be more familiar.

What we want students to recognize is that when two lines have different slopes, they will always intersect once! Which is how there is one solution. 

Infinite solutions

Lastly, you’ll want to talk about infinite solutions. From the graph, this is a bit more complicated for students to understand.

You’ll want to frame your questions differently so that you can communicate to students that there are two lines graphed one on top of the other. This means they have the same slope and `y`-intercept. Since they share all the same points, there will be infinite solutions!

Whiteboard game

Starting on slide `8`, you can begin a game with the class. If you have whiteboards, hand them out to the class. If you don’t have whiteboards, you can have students cut up a sheet of paper into `3`, then write infinite solutions on one, no solutions on another, and one solution on the last.

As each system is displayed on the board, have students come up with the number of solutions it would have. These are designed to be able to see just by a quick inspection of the equations, rather than actually graphing or solving them. When you sense a lull and it seems like students are finished, have them all show their answers! You can ask a student or multiple students to explain how they were able to figure it out.