Equations & Unknowns
Solving simple linear equations, finding the value of an unknown, and balancing equations.
What is Equations & Unknowns?
An equation is a mathematical statement that two things are equal. When an equation contains an unknown (usually represented by a letter like x or n), you need to find the value that makes the equation true.
In the 11+ exam, you will meet simple equations that can be solved using inverse operations. The key principle is: whatever you do to one side of the equation, you must do the same to the other side.
Step-by-Step Method
Identify what you need to find
Look at the equation and identify the unknown letter.
Use inverse operations to isolate the unknown
To undo addition, subtract. To undo multiplication, divide. Work backwards through the operations.
Do the same to both sides
Whatever you do to the left side, do exactly the same to the right side.
Simplify step by step
Work through one operation at a time until the unknown is on its own.
Check by substituting back
Put your answer back into the original equation to verify it works.
Worked Examples
Solve 3n + 7 = 22
Working
- Subtract 7 from both sides: 3n = 22 – 7 = 15.
- Divide both sides by 3: n = 15 / 3 = 5.
- Check: 3(5) + 7 = 15 + 7 = 22. Correct!
Solve 2x – 5 = 11
Working
- Add 5 to both sides: 2x = 11 + 5 = 16.
- Divide both sides by 2: x = 16 / 2 = 8.
- Check: 2(8) – 5 = 16 – 5 = 11. Correct!
Solve 4a + 3 = a + 15
Working
- Subtract a from both sides: 3a + 3 = 15.
- Subtract 3 from both sides: 3a = 12.
- Divide both sides by 3: a = 4.
- Check: 4(4) + 3 = 19, and 4 + 15 = 19. Correct!
Common Mistakes
Doing an operation to one side but forgetting to do it to the other.
Always write both sides of the equation on each line. Whatever you do to one side, do to the other.
Using the wrong inverse operation (e.g. subtracting instead of dividing).
Think about what was done to the unknown and undo it: addition undoes subtraction, multiplication undoes division.
Making sign errors with negative numbers.
Be extra careful with minus signs. Write each step clearly.
Top Tips
- Always check your answer by substituting it back into the original equation.
- If the unknown appears on both sides, collect the unknowns on one side first.
- Think of the equation as a balance scale – both sides must always stay equal.
- Undo operations in reverse order: undo addition/subtraction first, then multiplication/division.
Ready to practise?
Put these techniques into action with our free practice papers.
Practise Maths Questions