# A Student Shares a Great Observation

Today in my elementary algebra class we were reviewing solving systems of equations by addition or substitution. We were going over a system where students were having trouble determining what number to multiply each equation by in the system in order to eliminate the variable *x*. We had been discussing that the goal is to find the LCM of 12 and 14, but I told them that on exam day they could always fall back on multiplying each equation by the coefficient of *x* in the other equation (while making sure that produced one positive coefficient and one negative coefficient).

I then showed the class that they could find the LCM by finding the prime factorization of 12 and 14, gathering the results in a Venn diagram. Multiplying 6 by 2 by 7, the LCM is 84.

I finally showed them where the Venn diagram tells us that we can multiply 12 by 7 to get 84, and 14 by -6 to get -84.

One of my students pointed out to me (and the entire class) how he came to decide that he could multiply the two equations by 7 and -6. He started with 14 and -12, then divided both of those by their common factor of 2.

I loved the original thought, and repeated what he said to the entire class. I saw a lot of heads shaking in approval, and my students have a new strategy to use when the LCM does not jump out at them. It was a great day in elementary algebra!

*Do you have a story about a student discovery to share? I’d love to hear from you through the contact page on my website, posting a comment, or by reaching out to me on Twitter (@georgewoodbury).*

-George