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.

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.

My first through was this was another cool example of the power of brute force solving the underlying quartic rather than searching for factorizations but actually
its fun to actually rotate the equations back into the non-titled forms.

## Comments

I ended up riffing off this problem a bit (see the end of http://mymathclub.blogspot.com/2018/02/in-praise-of-rational-roots-theorem.html)

My first through was this was another cool example of the power of brute force solving the underlying quartic rather than searching for factorizations but actually

its fun to actually rotate the equations back into the non-titled forms.

Anyway it was fun to think about

Thanks