You can already solve equations with variables on both sides. This lesson is about why those moves are allowed. Once you know the reason, you never need a memorized trick like “move it across and flip the sign,” and you can tell at once when a step is wrong.
Why it works
An equation says two amounts are equal, like a level balance. If two amounts are equal and you do the same thing to both of them, they stay equal: add the same number, subtract the same number, multiply by the same number, or divide by the same nonzero number. Each of these moves can also be undone, so the new equation has exactly the same solutions as the old one. No solution is lost and none is added. That is the whole idea behind solving.
| Equation | What was done | Why it is allowed |
|---|---|---|
| $3x + 2 = x + 8$ | start | |
| $2x + 2 = 8$ | subtract $x$ from both sides | equals minus equals are equal |
| $2x = 6$ | subtract $2$ from both sides | equals minus equals are equal |
| $x = 3$ | divide both sides by $2$ | equals divided by the same nonzero number are equal |
Worked examples
Worked example: Giving a reason for each step
Solve $5x - 7 = 2x + 8$ and justify each step.
- Subtract $2x$ from both sides: $3x - 7 = 8$. (Same amount from both sides.)
- Add $7$ to both sides: $3x = 15$. (Same amount to both sides.)
- Divide both sides by $3$: $x = 5$. (Same nonzero divisor on both sides.)
- Check: $5(5) - 7 = 18$ and $2(5) + 8 = 18$.
Answer: $x = 5$
Worked example: Rewriting one side is different
Solve $4(x - 2) = 2x + 6$.
- Distribute on the left: $4x - 8 = 2x + 6$. This only rewrites the left side in an equal form, so the right side does not change.
- Subtract $2x$ from both sides: $2x - 8 = 6$.
- Add $8$ to both sides: $2x = 14$.
- Divide both sides by $2$: $x = 7$. Check: $4(5) = 20$ and $14 + 6 = 20$.
Answer: $x = 7$
Common mistake
Using “move a term across and flip its sign” on something that is not a term. In $3(x + 4) = 21$, the $4$ is inside a product, so “moving it” to get $3x = 21 - 4$ is wrong. The left side is $3$ times $(x + 4)$; undo the multiplication first ($x + 4 = 7$) or distribute first ($3x + 12 = 21$). Check any answer by putting it back into the original equation.
The one move that is never allowed is multiplying both sides by $0$: it turns every equation into $0 = 0$ and cannot be undone.
Key idea
Every step must do the same thing to both whole sides (add, subtract, multiply, or divide by a nonzero number), or rewrite one side in an equal form. Then each new equation has exactly the same solutions as the one before.