# Solving linear equations by factoring

I usually teach solving linear equations by balancing both sides e.g.

$$\begin{array}{cccc} 2x&+&3&=&5\\ &&\color{red}{-3}&&\color{red}{-3}\\ 2x&&&=&2\\ \color{red}{\div2}&&&&\color{red}{\div2}\\x&&&=&1 \end{array}$$

Recently I have been considering teaching students to solve via factoring $$\begin{array}{cccc} 2x&+&3&=&5\\ &&\color{red}{-5}&&\color{red}{-5}\\ 2x&-&2&=&0\\ 2(x&-&1)&=&0\\x&&&=&1 \end{array}$$

I think an advantage might be that solving quadratic equations is more aligned with previous learning. However I can see students being confused by factoring.

Has anybody taught solving linear equations by factoring? What were the outcomes for the students and future progress?

Any thoughts would be appreciated.

• I have no answer because I would never have thought to do this, but that makes me even more interested in seeing answers here. Is this effective? How and why? – Brendan W. Sullivan May 3 '15 at 2:11
• Perhaps I'm too detached from a learner's way of thinking, but is there really a difference for them? There hardly is any for me. – Git Gud May 3 '15 at 12:06
• @gitgud in my experience they are very different for students. Different still is using inverse functions $5\implies -3\implies \div2 \implies 1$ they tend to find this easiest to understand. (Ed: fixed MJ, it is hoped, quid) – Karl May 3 '15 at 12:50
• To clarify the last step is the zero factor property not division by 2 – Karl May 3 '15 at 12:59
• Absolutely, to students there would be a huge difference between the two methods. My authority comes from working with tons of high school and early college students. The important criterion will be long term progress. – nickalh May 9 '15 at 8:25

## 1 Answer

I think the reason for students' confusion in the latter method depends on two parameters (assuming the students are aware of both concepts):

1. In which order are the two concepts taught?
2. How long was the gap between teaching both concepts?

It all boils down to the students' reluctance in connecting two different concepts that they were taught. Students should be trained as mathematical thinkers -- problem solvers. This attitude will help them become at ease in applying two concepts together to solve a problem.

So your method indeed reinforces this attitude. Perhaps your students are not yet acquainted with such disposition. However, I would highly recommend the second method (and similar out of box techniques to solve seemingly mundane problems).