Math 7 · Solving equations
Do the same to both sides, and undo in reverse
An equation is a balance: the left side is worth exactly what the right side is worth. Solving one means getting the variable alone without tipping the balance — whatever you do to one side, you do to the other — and then checking the answer by putting it back.
- 1. The words, first
- 2. Keeping an equation balanced
- 3. One-step and two-step equations
- 4. Writing equations for problems
- 5. What costs marks
The words, first
The idea: Preservation of equality is a long name for a short rule: the same to both sides.
| Word | What it means |
|---|---|
| Equation | Two expressions joined by an equals sign, such as x + 5 = 12. The equals sign means “has the same value as”. |
| Variable | A letter standing for an unknown number, like x. |
| Side | Everything to the left of the equals sign, or everything to the right. |
| Solution | The value of the variable that makes the equation true. For x + 5 = 12 it is x = 7. |
| Preservation of equality | Doing the same to both sides keeps an equation true: add, subtract, multiply or divide both sides by the same number (never divide by 0). |
| Inverse operation | The operation that undoes another. Subtracting undoes adding; dividing undoes multiplying. |
| Zero pair | +1 and −1 together, worth 0. Adding −6 to +6 makes six zero pairs. |
| Verify | Substitute the solution into the original equation and check that both sides come out equal. |
Keeping an equation balanced
The idea: Picture a level pan balance. Taking the same amount off both pans, or adding the same to both, keeps it level.
x + 5 = 12 as a balance. A bag holding an unknown number of blocks, plus 5 loose blocks, balances 12 blocks. Take 5 blocks off each pan. It is still level, and the bag alone balances 7 blocks: x = 7.
The equals sign is not “the answer is”. In 3 + 4 = □ + 2, the box is 5, not 7. The left side is worth 7, so the right side must be too.
With integers, use zero pairs. To solve x + 6 = 2, add −6 to both sides. On the left, +6 and −6 make six zero pairs and vanish. On the right, 2 + (−6) = −4. So x = −4. Verify: −4 + 6 = 2 ✓.
Subtraction works the same way. For x − 3 = −8, add 3 to both sides: x = −8 + 3 = −5. Verify: −5 − 3 = −8 ✓.
“Move the number across and change its sign” is a shortcut for exactly this: adding the opposite to both sides.
One-step and two-step equations
The idea: Undo what was done to the variable, in reverse order: undo adding or subtracting first, then undo multiplying or dividing.
| Equation | What was done to x | Undo it on both sides | Solution |
|---|---|---|---|
| 3x = 21 | multiplied by 3 | divide by 3 | x = 7 |
| x/4 = 6 | divided by 4 | multiply by 4 | x = 24 |
| x + 9 = 4 | 9 added | subtract 9 | x = −5 |
Why reverse order. Socks go on before shoes, so shoes come off first. The last thing done to x is the first thing to undo.
Worked example: 5x + 3 = 38.
- What happened to x: multiplied by 5, then 3 added.
- Undo the adding first. Subtract 3 from both sides: 5x = 35.
- Then the multiplying. Divide both sides by 5: x = 7.
- Verify: 5 × 7 + 3 = 35 + 3 = 38 ✓.
The wrong order, caught. Dividing first gives 38 ÷ 5 = 7.6, then 7.6 − 3 = 4.6. Verifying shows the problem: 5 × 4.6 + 3 = 26, not 38. The 3 was never divided.
Writing equations for problems
The idea: Say what the variable stands for, write one equation that says what the story says, solve it, check it in the story, and answer in a sentence with units.
Worked example. A movie club costs $8 to join, then $6 for each movie. Ella has spent $44 in all. How many movies has she seen?
- Name the variable: let m be the number of movies. The cost is 6m + 8.
- Write the equation: 6m + 8 = 44.
- Solve: subtract 8 from both sides, 6m = 36. Divide both sides by 6, m = 6.
- Check in the story: 6 movies at $6 is $36, and the $8 fee makes $44 ✓.
- Answer: Ella has seen 6 movies.
Dividing in the story. Four friends split a bill evenly and each pays $13. With b for the bill, b/4 = 13. Multiply both sides by 4: the bill was $52.
Integers in the story. The temperature rose 9 degrees and is now 4 °C. What was it before? t + 9 = 4, so t = 4 − 9 = −5 °C. Check: −5 + 9 = 4 ✓.
What costs marks
The idea: Verifying catches the first three.
- Doing something to one side only, or to only part of a side.
- Undoing in the wrong order. In 5x + 3 = 38, subtract the 3 before dividing by 5.
- Sign slips with integers. x + 6 = 2 gives x = −4, not 4 or 8.
- Not verifying. Substituting takes seconds.
- Stopping at m = 6. Answer the question in words.