Solving by adding or subtracting

Take 5 away from both pans of a level balance and it stays level. That one rule turns x + 5 = 12 into x = 7.

An equation as a balance

Picture a pan balance with a closed box and five unit weights on the left, and twelve unit weights on the right. The pans are level, so the contents weigh the same. That is the equation x + 5 = 12, where x is the weight in the box. You cannot open the box, but you can take weights off. If you remove five weights from the left pan, you must remove five from the right pan too, or the balance tips. After that the box sits alone on the left with seven weights on the right, and the balance is still level. So x = 7.

The balance picture is the Subtraction Property of Equality. In words: if two quantities are equal and you subtract the same number from both, the results are still equal. In symbols, if a = b then a − c = b − c. The Addition Property of Equality works the same way in the other direction: if a = b then a + c = b + c. OpenStax uses these two properties to solve every one-step equation of the form x + a = b or x − a = b. The goal is always the same, to get the variable by itself on one side.

Undo what was done to the variable

To decide which property to use, read what has been done to the variable and do the opposite. In y + 37 = −13, 37 has been added to y, so subtract 37 from both sides: y = −13 − 37 = −50. In a − 28 = 64, 28 has been subtracted from a, so add 28 to both sides: a = 64 + 28 = 92. Addition and subtraction undo each other, which is why they are called inverse operations. Negative numbers follow the same rule. In x + (−4) = 9, the −4 is undone by adding 4, which gives x = 13.

The variable does not have to be on the left. In 15 = n + 6, subtract 6 from both sides to get 9 = n, which is the same statement as n = 9. Equality works in both directions, so you may swap the sides at the end if you like the variable on the left. Sometimes a side needs tidying first. In x − 3 + 5 = 10, combine −3 + 5 into +2, so the equation reads x + 2 = 10 and x = 8. Simplify each side as much as you can, then use one property to finish.

Check every answer in the original equation. For y + 37 = −13, put in −50: −50 + 37 = −13, which matches. The check takes a few seconds and catches the most common slip, adding when you should subtract. In the graph lab below, the left side x + 5 is the blue line and the right side 12 is the orange line. They meet above x = 7. Change d to 20 and see the solution move to 15.