What an equation says
An equation is a statement that two expressions are equal, such as x + 7 = 15. It contains an unknown (here x), and solving the equation means finding the value that makes the statement true. A linear equation is one where the unknown appears only to the first power, with no x², no x in a denominator and no products of unknowns. Examples are 3x − 5 = 16 and 2(x + 3) = 18.
The balance idea
The easiest picture is a set of scales in balance. The expression on the left weighs the same as the one on the right. If you add something to one side, the scales tip, unless you add the same to the other side. So the rule is:
Whatever you do to one side of an equation, do to the other side.
Doing the same thing to both sides keeps the equation true, and the aim is to end up with the unknown alone on one side: x = something.
Inverse operations
To get x on its own, undo what has been done to it, using inverse operations:
- Addition is undone by subtraction, and subtraction by addition.
- Multiplication is undone by division, and division by multiplication.
Example 1: x + 7 = 15
The 7 is added to x, so subtract 7 from both sides:
x + 7 − 7 = 15 − 7, so x = 8.
Check by putting 8 back in: 8 + 7 = 15. ✓
Example 2: 3x − 5 = 16
Two operations have been applied to x: it was multiplied by 3, then 5 was taken away. Undo them in reverse order, starting with the last one.
- Add 5 to both sides: 3x = 21.
- Divide both sides by 3: x = 7.
Check: 3 × 7 − 5 = 21 − 5 = 16. ✓
The working in reverse order matches the order of operations. Because multiplication happens before subtraction when evaluating 3x − 5, the subtraction is undone first.
Example 3: x/4 + 3 = 7
x was divided by 4, and then 3 was added.
- Subtract 3 from both sides: x/4 = 4.
- Multiply both sides by 4: x = 16.
Check: 16 ÷ 4 + 3 = 4 + 3 = 7. ✓
Example 4: equations with brackets, 2(x + 3) = 18
There are two ways to proceed.
Method A: divide first. Divide both sides by 2: x + 3 = 9. Subtract 3: x = 6.
Method B: expand first. 2x + 6 = 18. Subtract 6: 2x = 12. Divide by 2: x = 6.
Check: 2(6 + 3) = 2 × 9 = 18. ✓ Both routes give the same answer, and it is fine to choose whichever feels easier. Dividing first works well when the number outside divides evenly into the other side.
Example 5: unknown on both sides, 5x + 2 = 3x + 10
Collect the x terms on one side and the numbers on the other.
- Subtract 3x from both sides: 2x + 2 = 10.
- Subtract 2 from both sides: 2x = 8.
- Divide by 2: x = 4.
Check both sides: the left is 5 × 4 + 2 = 22, and the right is 3 × 4 + 10 = 22. ✓
A tip is to move the smaller x term, so that the coefficient of x stays positive. It is not required, but it reduces sign mistakes.
Setting up from words
Many practical problems start in words. “I think of a number, double it and add 3, and the result is 11.” Let n be the number. Then 2n + 3 = 11. Subtract 3: 2n = 8. Divide by 2: n = 4. Check: 2 × 4 + 3 = 11. ✓
Choosing a letter and writing what it stands for is the first step, and it uses the ideas in algebra basics. If you are comfortable with that, the hardest part is translating the sentence, so go word by word.
Special cases
Not every equation has exactly one solution.
- No solution: 2x + 1 = 2x + 3. Subtracting 2x from both sides gives 1 = 3, which is false. No value of x works.
- Every number works: 2x + 1 = 2x + 1. Both sides are always the same, so the equation is true for any x.
If you reach a statement that has no x left in it, look at whether it is true or false. This tells you which case you are in.
Common mistakes
- Doing an operation to only one side.
- Subtracting when you should add. To undo “−5”, add 5.
- Forgetting to apply the operation to every term. If you multiply both sides by 4, every term on both sides is multiplied.
- Dropping a negative sign when moving terms.
- Not checking the answer. A check takes a few seconds and catches most errors.
When something goes wrong, work backwards through your steps line by line, in the same way you would approach a debugging mindset in code: find the first line that is wrong, not the last.
How to practice
Studying worked examples and then solving similar problems with a gradually reduced amount of help is a common way to learn this topic, and the approach is often described as fading the support gradually. Try covering the solution to a worked example and reproducing it from memory, which is a form of retrieval practice. Mixed practice, where equation types are shuffled rather than grouped, can also help you learn to choose a method, though results vary between learners.
Equations are used again in other topics. Finding a percentage, converting units and scaling in ratios and proportions all rely on being able to isolate an unknown quantity.