Solving equations systematically: transposing

The balance method works perfectly, but it is wordy. After a few problems you start to notice a pattern: each "add to both sides" effectively moves a term from one side to the other, flipping its sign. That shortcut is called transposing.

Idea

Transposition rules.

1. A term added on one side becomes subtracted on the other (and vice versa).

   $2y + 7 = 21 \quad\Longrightarrow\quad 2y = 21 - 7.$

2. A factor that multiplies on one side becomes a divisor on the other (and vice versa).

   $2y = 14 \quad\Longrightarrow\quad y = \frac{14}{2}.$

3. A divisor on one side becomes a multiplier on the other.

   $\frac{u}{15} = 6 \quad\Longrightarrow\quad u = 6 \times 15.$

These shortcuts are exactly the balance moves, just written more compactly. They do not change what you do , only how much you write.

Standard procedure. To solve an equation of the form $ax + b = c$ or $ax + b = dx + e$:

1. Bring the unknowns to one side, numbers to the other. Transpose $dx$ from the right to the left (or vice versa, whichever leaves a positive coefficient).
2. Combine like terms on each side. You should be left with $(\text{number})\cdot x = \text{number}$.
3. Divide both sides by the coefficient of $x$.

Check by substitution. Always plug your answer back into the original equation. It is the single most reliable way to catch sign errors.

Equations with no solution / infinitely many. Sometimes transposing eliminates the variable entirely:

• $x + 5 = x + 4$ becomes $5 = 4$ , never true. No solution.
• $2(x + 3) = 2x + 6$ becomes $0 = 0$ , always true. Every value of $x$ works.

Both cases tell you something about the problem: the two expressions are either incompatible, or they are the same expression in disguise.

Worked examples

Example 1. Solve $4(m + 6) - 8 = 2m - 4$.

Open the bracket: $4m + 24 - 8 = 2m - 4$, i.e. $4m + 16 = 2m - 4$.

Transpose $2m$ to the left and $16$ to the right: $4m - 2m = -4 - 16$, so $2m = -20$, hence $m = -10$.

Check: LHS $= 4(-10 + 6) - 8 = 4(-4) - 8 = -24$. RHS $= 2(-10) - 4 = -24$. ✓

Example 2. Solve $5s = 3s + 12$.

Transpose $3s$: $5s - 3s = 12$, so $2s = 12$, hence $s = 6$. Check: LHS $= 30$; RHS $= 18 + 12 = 30$. ✓

Example 3. Solve $3u - 7 = 2u + 3$.

Transpose: $3u - 2u = 3 + 7$, so $u = 10$. Check: $3(10) - 7 = 23$; $2(10) + 3 = 23$. ✓

Example 4. Frame and solve: an equation that has no solution.

Try "$x + 4 = x + 5$." Transposing $x$: $4 = 5$. False. So no value of $x$ works. This makes sense , you cannot have a number whose $4$-more equals its $5$-more.

Try it yourself

1. Solve $3x - 10 = 35$.
2. Solve $5s = 3s + 14$.
3. Solve $3u - 7 = 2u + 3$.
4. Solve $4(m + 6) - 8 = 2m - 4$.
5. Solve $\dfrac{u}{15} = 6$.
6. Solve $7x + 5 = 2x + 30$.
7. Solve $2(y - 3) = y + 5$.
8. Frame an equation that has no solution and explain why.

Activity

Pick a number. Multiply it by $3$, subtract $4$, then double the result. Tell a friend only the final answer. Can your friend find your original number? They will set up an equation $2(3x - 4) = \text{(your answer)}$ and solve it. Try this five times with different numbers.