Solving One-, Two- and Multi-step Equations

An equation says two things are equal, like a set of balance scales. x + 7 = 19 means the left pan and the right pan weigh the same. To find x you have to get it alone on one side — and the only rule is that whatever you do to one side, you must do to the other, or the scales tip and the equation stops being true. Here 7 is being added to x, so you subtract 7 from both sides: x = 12. Then check by putting your answer back into the original: 12 + 7 = 19 ✓. That check takes five seconds and catches almost every mistake you will make.

Every operation has an opposite that undoes it. Addition is undone by subtraction, multiplication by division, and the other way round in each case. So the question to ask is always: what is being done to x? In 3x = 21, x is being multiplied by 3, so you divide both sides by 3 to get x = 7. The classic error is subtracting 3 instead, which would give 18 — but 3 was never added to x, it was multiplied, so subtracting undoes the wrong thing.

A two-step equation has two things done to x, so you undo them both — in reverse order, like taking off a jumper before a shirt. In 2x + 5 = 17, x was multiplied by 2 and then 5 was added. So you undo the addition first: subtract 5 from both sides to get 2x = 12. Then undo the multiplication: divide both sides by 2 to get x = 6. Doing it in the other order — dividing first — is much messier, because you would have to divide the 5 as well. Always strip away the term that is furthest from x first.

A fraction bar is just a division sign, so undo it by multiplying. In x over 3, plus 2, equals 7: first subtract 2 from both sides, leaving x over 3 equals 5. Then multiply both sides by 3 to get x = 15. Check: 15 divided by 3 is 5, plus 2 is 7 ✓. When you multiply, multiply the whole side — if the equation had been x over 3 plus 2 equals 7 and you multiplied straight away without subtracting first, you would need to multiply the 2 and the 7 as well, which is why subtracting first is simpler.

Set your working out one line per step, with the equals signs lined up underneath each other. It looks tidy, but the real reason is practical: if you make a slip, you and the marker can see exactly which line it happened on, and you keep the marks for everything before it. A bare answer with no working scores nothing if it is wrong. Finish by substituting your answer back into the original equation — 4 times 8 minus 9 gives 23 ✓. That habit will catch nearly every error you make in this chapter.