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Algebra

A two-step equation is two things done to an unknown, and solving it is undoing them backwards. That is the whole method, and it is why the order matters so much: whatever was done LAST has to be undone FIRST, the same way you take your coat off before your jumper.

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Two steps, no bracket

3n + 2 = 23

  1. Read what was done to n: it was multiplied by 3, and then 2 was added.
  2. The last thing done was adding 2, so that comes off first: 23 - 2 = 21.
  3. Now 3n = 21, and the multiplication is undone by dividing: 21 divided by 3.
  4. n = 7. Check it: 3 times 7 is 21, add 2 is 23.

Worth knowing. Subtract before you divide, because the addition happened last. Dividing 23 by 3 first gives a number that is not whole, which is often the first clue that the order has gone wrong.

Two steps with a bracket

3(n + 2) = 15

  1. The bracket says the adding happened FIRST this time, and the multiplying second.
  2. So undo the multiplying first: 15 divided by 3 is 5.
  3. Now n + 2 = 5, and the 2 comes off: 5 - 2.
  4. n = 3. Check it: 3 plus 2 is 5, times 3 is 15.

Worth knowing. This equation and the one above look almost identical and are undone in opposite orders. The bracket is the only thing telling you which is which, which is why it is worth reading before you start rather than after.

Where it usually goes wrong

  • Undoing the steps in the order they are written instead of backwards — dividing before subtracting on 3n + 2 = 23.
  • Ignoring the bracket, so 3(n + 2) = 15 gets solved as though it were 3n + 2 = 15.
  • Undoing only the first step and stopping. On 3n + 2 = 23 that gives 21, which is what 3n is worth and not what n is worth.

The Professor reads the equation out loud as a story before he solves it. "Start with n, times 3, then add 2" — and then he tells it backwards. He has never once got the order wrong since he started doing that.

Practise it

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Questions teachers and parents ask

Why do I undo the steps backwards?

Because that is the only order that unwraps them. If someone puts a letter in an envelope and then puts the envelope in a box, you open the box first. Doing it the other way round is not possible, and the same is true of an equation.

How do I know whether there is a bracket?

It is written. 3n + 2 means the multiplying happened first and 3(n + 2) means the adding did, and the two have different answers for the same numbers. Reading the equation aloud as a sequence of steps before solving it makes the difference obvious.

What if the division does not come out whole?

On this site it always will, because the equations are built from a whole answer outwards. If you get a fraction while solving one, it almost always means two steps were undone in the wrong order — which makes it a useful warning rather than a dead end.

When are two-step equations taught?

Grade 7 (7.EE.B.4), which asks for both the px + q = r form and the p(x + q) = r form. One-step equations come first, in grade 6, and they are in the pre-algebra topic.

Where this shows up

Standards

  • 6.EE · Expressions and equations
  • 7.EE · Expressions and equations
  • CCSS.MATH.CONTENT.7.EE.B.4 — Use variables to represent quantities and construct simple equations to solve problems, solving equations of the form px plus q equals r and p times the quantity x plus q equals r.
  • CCSS.MATH.CONTENT.6.EE.B.6 — Use variables to represent numbers and write expressions when solving a problem, understanding that a letter can stand for any number in its set.

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