Linear Equations
An equation is a balanced scale. On each pan sit some bags (each holding the same unknown number of weights) and some loose weights. Solving means finding out what's in a bag, and there is only one rule: whatever you do to one side, do to the other.
National curriculum KS3: use algebraic methods to solve linear equations in one variable, including all forms that require rearrangement, with the unknown on one side or both sides1 · What an equation says
x + 3 = 7 is a statement: "some number, plus three, makes seven". The = sign is not an instruction to work something out; it's a claim that the two sides weigh the same. An expression (like x + 3) is just one pan; an equation has two pans and an = between them.
Put the equation on a scale. Teal bags hold x weights each (we don't know how many). Orange discs are single weights. If the scale balances, the equation is true. Drag the slider to try values of x and see which one balances.
Trying values works for small numbers, but it's guessing. The rest of this page is the method that always works.
2 · The one rule: keep it balanced
If the scale balances and you take 3 weights off the left only, it tips. Take 3 off both sides and it still balances, but now the left pan is simpler. Repeat until a single bag sits alone on one side. Whatever is on the other side is what's in the bag.
Solve this one yourself. Click an operation; it's applied to both pans at once. Try to reach x = … in as few moves as you can.
Why not just "move it across and change the sign"? That shortcut is a description of subtracting from both sides, and it works, but students who only know the shortcut get lost when the unknown is on both sides or when there's a division. Think "both sides" and the shortcut becomes obvious instead of magic.
3 · Inverse operations: undo in reverse order
Another way to see it. 2x + 3 = 11 was built by taking x, multiplying by 2, then adding 3. To get x back, undo the operations in reverse order: subtract 3 first, then divide by 2. Like taking off shoes then socks: last on, first off.
4 · Unknowns on both sides
When bags appear on both pans, the extra move is to remove the same number of bags from each side. Take away as many as the side with fewer bags has, so that bags remain on only one pan. Then it's a one-sided equation and you finish as before.
A bar model shows it a different way. Two bars of equal length, one for each side. Line them up: the parts that match can be removed from both, and what's left must also match.
Which side should the x end up on? Either, but life is easier if you remove the smaller number of bags so the remaining coefficient is positive. In 2x + 9 = 5x + 3, take 2x from both sides (leaving 9 = 3x + 3) rather than 5x (leaving −3x + 9 = 3). Both are correct; one avoids negatives.
5 · Brackets, fractions and negatives
Three extras that appear in "all forms requiring rearrangement". Each is one preparatory step before the balance method takes over.
| Form | First move | Then |
|---|---|---|
| 3(x + 2) = 18 | Expand: 3x + 6 = 18. (Or divide both sides by 3 first: x + 2 = 6.) | x = 4 |
| x/4 − 1 = 2 | + 1 both sides: x/4 = 3. Then × 4 both sides. | x = 12 |
| (x + 5)/2 = 7 | × 2 both sides to clear the fraction: x + 5 = 14. | x = 9 |
| 7 − 2x = 1 | − 7 both sides: −2x = −6. Then ÷ (−2). | x = 3 |
| 12 = 3x − 6 | Unknown on the right is fine. + 6: 18 = 3x. ÷ 3. | 6 = x, i.e. x = 6 |
Type any linear equation and the solver shows every both-sides step, with a substitution check at the end.
6 · Always check
Substitute your answer back into the original equation. If both sides come out the same, you're right. This takes ten seconds and turns a hopeful answer into a certain one. Every tool on this page ends with this step for a reason.
7 · Where equations come from
Number puzzles
"I think of a number, treble it, subtract 4 and get 17." → 3x − 4 = 17 → x = 7. The equation is the puzzle written down.
Geometry
Angles in a triangle: x + 2x + 60 = 180 → 3x = 120 → x = 40. Perimeter 2(x + 3) + 2x = 30 → x = 6. Shapes generate equations constantly.
Money
Two phone tariffs: 10 + 0.05m and 4 + 0.08m. When do they cost the same? 10 + 0.05m = 4 + 0.08m → m = 200 minutes. Unknown on both sides, straight from real life.
Science
v = u + at with v = 30, u = 6, a = 4: 30 = 6 + 4t → t = 6 s. Every "find the time" question is an equation.
Programming
The computer can't rearrange, so you solve total = 25 + 0.12*m for m before writing m = (total - 25)/0.12. Rearranging a formula is solving an equation with letters instead of numbers.
Break-even
Cost 500 + 3n, income 8n. Break even when 500 + 3n = 8n → n = 100 items. Businesses solve this before they start.
Worked examples
Solve 4x − 7 = 13
Solve 5x + 2 = 2x + 11
Solve 3(2x − 1) = 4x + 9
Exam-style: the angles of a triangle are x, x + 20 and 2x − 40 degrees. Find the largest angle.
Quick check
What you've learned
An equation is a balance; solving it means isolating one bag. The only rule is to do the same thing to both sides, and the strategy is to undo operations in reverse order: deal with added or subtracted numbers first, then divide by the coefficient. With unknowns on both sides, first remove the smaller number of bags from each pan. Brackets get expanded, fractions get multiplied away, and a negative coefficient is just one more division. Then substitute back and check.