Monday, March 12, 2012

Substitution with Some Flair

Some of my students who prefer not to solve systems by substitution have come up with a new way of substitution that I actually REALLY like. I am going to share it here with you.

Say the problem indicates to solve this following system using substitution (I know, it's totally artificial to prescribe a specific method, but for shared assessment reasons, I want them to still be prepared for questions like this on the semester exam):

2x + 3y = 48
3x - 4y = 4

Basically, some of the kids who dislike substitution have invented a hybrid of the two methods of substitution and elimination. They first scale the equations to get matching terms:

(2x + 3y = 48)*3 becomes 6x + 9y = 144
(3x - 4y = 4)*2 becomes 6x - 8y = 8

If you solve the first equation for 6x, you get 6x = -9y + 144. Then, substitute this into the second equation, you get: (-9y + 144) - 8y = 8. Ingenious! I was impressed that they just invented this hybrid method. It's much easier than getting x = -3y/2 + 24 and plugging that into the second equation to get 3(-3y/2 + 24) - 4y = 4, because the hybrid method they have invented bypasses all of the fractions immediately and still satisfies the problem requirement of applying the concept / skill of substitution. Their method of substitution is so much more elegant than our traditional substitution.

LOVE! Amazing what kids can come up with all by themselves.

I am simply bubbling with excitement to finally start new topics this week! My short attention span really gets the better of me.


  1. I love it, too! It's amazing what can come out of trying to avoid fractions. :) Well done on your students' part.

  2. but if you subtract the second equation from the first one, you get
    17y=136 then divide by 17
    you get y=8.
    This way is even easier.

  3. Yeah they're pros at that, obviously, but that's just straight elimination. My colleagues would not accept that on an exam (shared assessment) as the kids demonstrating that they understand both methods...