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# “Are there systems that have three solutions?”

Students were prompted to graph a systems that has more than one solution, and one group provided the work above, confident that they had a system with three solutions.

So, how do you respond to the group? What do you say?

Thanks again to Nicole Paris for the submission.

## 2 replies on ““Are there systems that have three solutions?””

First, I absolutely love that this student has connected an intersection of two lines at a point as a means of representing a solution to a system of linear equations.

Next, I love the fact that they drew another line, and proved the important theorem that in a plane determined by two intersecting lines, a third line (that is non-collinear with the other two, since that’s not interesting) must either intersect with the other lines at 2 and only 2 points or…

[So in the spirit of Dan Meyer, that’s the initial dilemma of a three-act lesson following this up, how should I finish that sentence?]

I try to teach my geometry students to see things moving (in their heads), and see which angle is growing, versus which angle is decreasing. It’s intuition, but I’m hoping the dilemma above helps shove students toward lines that all intersect at one point, since that is a valid alternative to the triangle of three points above, isn’t it? So what would *that* one point mean? And how could you represent it?

I think this group has produced an excellent response, In some sense, this system of equations does have three solutions. The teacher is now perfectly positioned to challenge this group to explain what it means for a system of equations to be simultaneously satisfied.