Seeing ideas about substitution for the first time

My son had an interesting problem on his enrichment math homework this week, and it gave him a lot of trouble this morning:

Interesting problem on my son's enrichment math homework. Solve { x^2 – xy + y^2 = 13 and x – xy + y = -5 }. The section is on solving quadratics. I wonder if the idea they are trying to teach with this problem is u-substitution (u = x + y, v = xy) or something else.

— Mike Lawler (@mikeandallie) March 29, 2018

Tonight I thought it would be good to talk through the problem since I think the main idea he needed to solve it was new to him.

Here’s the introduction and some of the ideas he tried this morning:

Next we took a look at the equations on the computer and talked about some of the ideas we saw:

After looking at the graphs of the equations on the computer we came back to the whiteboard to talk about substitution.

Finally, having worked through the introductory part of u-substitution in the last video, I let him finish off the project on his own.

I can’t remember talking through this topic previously, but it was fun. It is always neat to be there when a kid is seeing a math topic for the first time.