Solving step by step

Now use these ideas on an equation with an unknown: 5x − 4 = 7. Trial and error would struggle here, because the answer is not a whole number.

What can we do so that only 5x is on one side and the equality still holds? To retain only 5x on the LHS, we must remove the term −4. That is done by ADDING 4 to both sides: 5x − 4 + 4 = 7 + 4, so 5x = 11.

Now to retain only the unknown x, we must remove the FACTOR 5. That is done by dividing both sides by 5: 5x ÷ 5 = 11 ÷ 5, so x = 11/5.

Can we be sure? Yes — CHECK IT. Substitute x = 11/5 back into the original equation and see whether LHS = RHS. LHS = 5(11/5) − 4 = 11 − 4 = 7, and the RHS is 7. They agree, so 11/5 really is the solution.

Example 5: solve 11y + (−5) = 61. To keep only the unknown term 11y on one side, remove the term −5, by subtracting (−5) from both sides: 11y + (−5) − (−5) = 61 − (−5), that is, 11y = 66.

From 11y = 66 you may simply see that 11 × 6 = 66, so y = 6. Or divide both sides by 11: 11y ÷ 11 = 66 ÷ 11, giving y = 6. Both routes are fine; the second always works, the first only when you happen to spot it.

Two moves, then, in this order: clear the TERMS first, so that only the unknown's term is left, then clear the FACTOR. And then check. Remember, it is always good to check your solution — substituting takes a moment and catches a slip that would otherwise travel all the way to the answer.