Back to Basics: Equation Manipulation

I am posting this because I actually witnessed the following dialogue take place in a classroom:

Teacher: To solve 7x + 3 = 12, first subtract 3 from both sides to get 7x = 9.  Next, divide by 7 on both sides to get x = 9/7.

Student: Can you start by dividing by 7 on both sides first?

Teacher: No, you have to subtract 3 first.


The Truth: If you apply the same function to equal quantities, you will get equal quantities.

So, yes, you can definitely start by dividing both sides by 7. You’d get x + 3/7 = 12/7, and now you can subtract by 3/7 on both sides to solve for x.

There’s a tremendous variety of functions, so there’s a great many ways of solving equations.  Keep this in mind if you’re just beginning your study of equation solving. Because there are generally many ways to solve an equation, don’t waste efforts trying to memorize specific derivations. Instead, focus on what a manipulation accomplishes and whether it moves toward the desired goal. Just for fun, you might even challenge yourself to come up with multiple derivations.

This particular teacher, in this particular case, is wrong. Nobody is always right, so don’t just trust everything you hear. Think about everything you learn at school and decide for yourself whether it rings true or not. If you’re still unsure about whether a mathematical concept is correct or not, feel free to ask us at


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3 Responses to Back to Basics: Equation Manipulation

  1. David says:

    One good solution to this recurring problem is to make sure teachers are taught well first and know precisely what they are doing and why. I don’t think we should just dismiss this as an instance of getting something wrong, because there are two kinds, one where we know that it is wrong but was just careless, the other where we don’t even know that it is wrong. I think that teacher falls into the second kind, which can never happen if we fully understand mathematical logic.

  2. Deleance Blakes says:

    Reblogged this on thought academy and commented:
    Very true. So many students now are–in my opinion–duped into thinking one mechanical way! There’s a lot of intuition being lied to out there!

  3. zacharybruce says:

    I had a friend in high school who failed a test for using l’hopital’s rule for finding rational limits. When he asked why it was marked wrong, the response was “I haven’t taught you that, so you must have been cheating.”

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