Unknowns on both sides

Example 6: solve 6y + 7 = 4y + 21. Something new here — expressions with an unknown stand on BOTH sides. We only know how to finish when the unknown is on one side, so the first job is to bring the unknown terms together.

Subtract 4y from both sides: 6y + 7 − 4y = 4y + 21 − 4y, which gives 2y + 7 = 21. Now it is an equation of the familiar kind. Subtract 7 from both sides: 2y = 14. Then divide both sides by 2: y = 7.

The procedure can be made more efficient with an observation. When we solved 11y + (−5) = 61, the first step was written 11y + (−5) − (−5) = 61 − (−5). But since that action simply REMOVES the term −5 from the LHS, we could have written it straight as 11y = 61 − (−5).

The same for the factor: instead of writing 11y ÷ 11 = 66 ÷ 11, we can write y = 66 ÷ 11 at once. Both shortcuts are just the inverse-operation idea, recorded in one line instead of two.

That gives three rules — sometimes called transposing (moving a term to the other side): (a) When a term added or subtracted on one side is removed from that side, its additive inverse should appear as a term on the other side. So 2y + 7 = 21 becomes 2y = 21 − 7.

(b) If one side is the PRODUCT of two or more numbers or expressions and we remove one of the factors, the other side should be DIVIDED by that factor. So 2y = 14 becomes y = 14 ÷ 2.

(c) If one side is the QUOTIENT of two numbers or expressions and we remove the divisor, the other side should be MULTIPLIED by that divisor. So u/15 = 6 becomes u = 6 × 15, giving u = 90. These three lines are the whole systematic method, and they are worth knowing by heart.

A quick memory line for moving across the equals sign: + becomes −, − becomes +, × becomes ÷, ÷ becomes ×. It is not a new trick — it is just doing the same operation to both sides, written in one step.