Solving grade 8 algebra... the Enterprise way

I have been tutoring grade 8 math for some time. And let me tell you that it is not easy. Doing grade 8 math is one thing, but tutoring/teaching it is a completely different story. The trouble is that, by dint of having done a lot of much harder math problems as a university graduate, high school math just seems to be second nature and it has become hard to recall or even explain the thinking process behind solving simple problems.

As someone who reads The Daily WTF religiously everyday, I thought I would be less likely to fall prey to Enterprise-hype. How wrong I am! Anyway, last week I pulled a Daily WTF while tutoring math to a girl. It turns out that I showed an Enterprise solution to a grade 8 algebra problem. Here it is:

Tommy's father is 5 times as old as he is. His mother is 2 years younger than his father. Together, the sum of their ages is 75. How old is each of them?

Having dealt with multi-variable systems of linear equations for so long, my approach was "naturally" to identify the unknown variables and then to translate the sentences into equations:

Now we do some substitutions:

Finally, we solve for T, F, and M:

I thought that this method was already pretty dirty because I haven't even put the equations into matrix form and used the more formal linear algebra methods that I learned in college (or even the Gauss-Seidel method).

Yesterday I was tutoring the girl again. This is the first thing she said as I took out my pencil, eraser, etc: "I asked my teacher about it and she said your method is completely wrong. She said that you are teaching me stuff that I am supposed to learn only next year so she doesn't understand why you're trying to confuse me." I was pretty shocked and "pwned" was written all over my face. As it turns out, what the girl's teacher has taught (and was thus expecting) was actually even dirtier:

Notice that they don't bother with things such as identifying variables or substitutions. The whole problem was solved pretty much using only 1 variable from start to end. The reduction in complexity and in number of steps is pretty amusing. I think it was a good lesson (for me).