Keep it balanced
An equation is a balance: two sides that weigh the same. Whatever you do to one side, do to the other, and it stays balanced. That single rule solves every equation in Grade 9 — and knowing where it stops working separates a right answer from a confident wrong one.
Why think of an equation as a balance?
Equations exist to find a number you cannot see from facts you can: how many months until two gym plans cost the same, how wide a rectangle is when you know its perimeter, how much current flows in a circuit. You know what the unknown does; you want to know what it is.
The word algebra comes from exactly this. Around the year 820 in Baghdad, al-Khwarizmi wrote a book on solving equations whose title names two operations, al-jabr and al-muqabala — roughly, restoring and balancing: moving a subtracted amount to the other side, and cancelling equal amounts from both sides. The first became “algebra”.
The balance guarantees that each line of working is true if the line before it was. Do the same to both sides of a true equation and you get another true equation, until the last one says x = 5. It is also why checking works: put the answer back into the first line, and the two sides must come out the same.
Where it turns up
- Seesaws — a 30 kg child 2 m from the pivot balances a 40 kg child 1.5 m from it, because 30 × 2 = 40 × 1.5
- Science 9 chemistry — mass is conserved, so if 10.0 g of reactants make 6.4 g of one product, the other product is 3.6 g
- Science 9 electricity — Ohm's law, V = IR: a 12 V battery across a 4 Ω resistor gives 12 = 4I, so the current is 3 A
- Choosing a plan — a plan with a sign-up fee and a lower monthly price catches up with a no-fee plan in the month the two totals are equal
- 1. The words, first
- 2. The same to both sides
- 3. Checking by substitution
- 4. Where the balance tips: inequalities
- 5. Where the balance breaks: dividing by a variable
The words, first
The idea: An expression is something you simplify. An equation is something you solve. The difference is the equals sign.
| Word | What it means |
|---|---|
| Equation | Two expressions with an equals sign between them: a statement that they have the same value. |
| Expression | Mathematics with no equals sign, such as 3x + 7. You can simplify it, but there is nothing to solve. |
| Side | Everything to the left of the equals sign, or everything to the right of it. |
| Inverse operation | The operation that undoes another: + and − undo each other, and so do × and ÷. |
| Verify | Substitute the solution into the original equation and work out each side separately. |
| Inequality | Two expressions compared with <, >, ≤ or ≥ instead of =. More like a tilted balance than a level one. |
The same to both sides
The idea: Undo what was done to the variable, in reverse order, and do each step to the whole of both sides.
Worked example. 4(x − 3) + 5 = 25. The last thing done to x was adding 5, so undo that first:
4(x − 3) + 5 = 25 → 4(x − 3) = 20 → x − 3 = 5 → x = 8
Another route, same answer. Expand first: 4x − 12 + 5 = 25, so 4x − 7 = 25, 4x = 32 and x = 8. Any step that keeps the balance is allowed, so different routes must meet at the same solution. If yours do not, one has a mistake in it.
Fractions: multiply every term on both sides. For x/3 + 2 = x/2 − 1, multiply by 6, the lowest common denominator — every term, including the 2 and the 1:
2x + 12 = 3x − 6 → 18 = x
Multiplying only the fraction terms gives 2x + 2 = 3x − 1 and x = 3: a balance tipped by doing something to part of a side.
Checking by substitution
The idea: Put the answer into the original equation and work out each side on its own. If they match, the answer is right, however it was found.
Worked example. Someone solves 3(x + 2) = 5x − 4 like this:
3x + 2 = 5x − 4 → 6 = 2x → x = 3
Check in the original. Left side: 3(3 + 2) = 15. Right side: 5(3) − 4 = 11. They are not equal, and the first line is where it went wrong: the 3 never multiplied the 2. Done properly, 3x + 6 = 5x − 4, so 10 = 2x and x = 5. Left: 3(7) = 21. Right: 25 − 4 = 21 ✓.
Check the original, not your working. Substituting into a line you copied wrongly will happily confirm a wrong answer. In a word problem, check the story itself: if two gym plans are meant to cost the same after 5 months, work out both bills. That catches an equation set up wrongly in the first place, which the equation cannot.
Where the balance tips: inequalities
The idea: An inequality is a tilted balance. Adding, subtracting, and multiplying or dividing by a positive all keep the tilt. Multiplying or dividing by a negative reverses it.
Why a negative reverses it. 2 < 5, but −2 > −5. Multiplying both sides by −1 swaps which side is heavier.
Worked example. −2x + 3 > 11. Subtract 3 from both sides: −2x > 8. Divide both sides by −2 and reverse the sign: x < −4.
Checking an inequality takes two substitutions. The boundary should make the sides exactly equal: −2(−4) + 3 = 11 ✓. A value from your answer should make the inequality true: x = −5 gives 13, and 13 > 11 ✓. The first checks the number; the second checks the direction.
Or keep the variable positive. Add 2x to both sides instead: 3 > 11 + 2x, then −8 > 2x, so −4 > x — the same answer, with no reversal needed.
Where the balance breaks: dividing by a variable
The idea: You may multiply or divide both sides by anything except zero. A variable might be zero, so dividing by one can throw the answer away.
Worked example. Solve 5x = 2x. Dividing both sides by x gives 5 = 2, which is false, and suggests there is no solution. But x = 0 works: 5(0) = 2(0). Dividing by x assumed x was not 0 — and 0 was the answer. Subtract instead: 3x = 0, so x = 0.
When the variable is underneath, multiplying by it is safe, because it cannot be zero there anyway. In 12/x = 3, x = 0 would make the left side meaningless, so 12 = 3x and x = 4.
Why dividing by zero is never allowed. 0 × 5 = 0 × 2, but 5 is not 2. Dividing both sides by zero would “prove” they were equal, and the trick “proofs” that 1 = 2 almost always hide a division by zero somewhere. Math 20-1 makes this a habit: every rational expression there comes with a list of values the variable may not take.