Solving equations: from one step to unknowns on both sides

What does it mean to solve an equation?

To solve an equation is to find the number the letter stands for. An equation says two things are equal, so x + 7 = 12 says "some number add 7 makes 12". The solution is x = 5, because 5 + 7 = 12. Checking means putting the answer back in and seeing that both sides match.

An equation is different from an expression, which has no equals sign (like 2x + 1), and from a formula, which links several quantities and works for any values (like P = 2l + 2w). The algebra basics guide explains the words.

Children start on this at primary school. They use inverse operations to check calculations, and in Year 6 maths they write missing number problems with letters. At secondary school, the key stage 3 programme of study asks for algebraic methods to solve linear equations, including all the forms that need rearranging. Key stage 3 covers Years 7 to 9 together, and each school decides the order, so the school's scheme of work is the place to see when it comes up.

How to solve equations: keep both sides balanced

The rule for solving equations is to do the same thing to both sides. Think of a balance: if you take 3 away from one side, take 3 away from the other, and it stays level. Each step uses the inverse operation to undo something that has happened to x:

• If something has been added, subtract it.
• If something has been subtracted, add it.
• If x has been multiplied, divide.
• If x has been divided, multiply.

Worked example: one-step equations

• x + 7 = 12. Subtract 7 from both sides: x = 5.
• 3x = 21. 3x means 3 × x, so divide both sides by 3: x = 7, because 21 ÷ 3 = 7.

Solving linear equations with two steps

A linear equation has x on its own, with no x² or higher powers. Most have two steps: undo the adding or subtracting first, then the multiplying or dividing.

Worked example: a two-step equation

Solve 4x + 3 = 23.

1. Subtract 3 from both sides: 4x = 20.
2. Divide both sides by 4: x = 5.

Check: 4 × 5 = 20, and 20 + 3 = 23. Both sides match.

Answers aren't always positive whole numbers. For 2x + 9 = 3, subtract 9 from both sides to get 2x = −6, then divide by 2: x = −3. Check: 2 lots of −3 is −6, and −6 + 9 = 3. The negative numbers guide covers the signs.

How to simplify algebraic equations first

If one side has like terms, collect them before you start. In 2x + 3x + 1 = 21, the 2x and 3x make 5x, so the equation is 5x + 1 = 21. Subtract 1 from both sides to get 5x = 20, then divide both sides by 5 to get x = 4. Then check: 5 × 4 = 20, and 20 + 1 = 21.

Solving equations with brackets

To solve an equation with brackets, you can expand the brackets first, or, if the number outside divides the other side exactly, divide both sides by it.

Worked example: solving 3(x − 2) = 15

By expanding: 3(x − 2) is 3x − 6, so 3x − 6 = 15. Add 6 to both sides: 3x = 21. Divide by 3: x = 7.

By dividing: 15 ÷ 3 = 5, so dividing both sides by 3 gives x − 2 = 5. Add 2: x = 7.

Both ways give x = 7. Check: 7 − 2 = 5, and 3 × 5 = 15. The expanding brackets guide shows how to multiply out brackets of every kind.

Solving equations with unknowns on both sides

When x is on both sides, get all the x terms onto one side first, by taking away the smaller number of x from both sides. Then it's a two-step equation. GCSE content lists unknowns on both sides as something every student is assessed on.

Worked example: 5x + 4 = 2x + 19

1. Subtract 2x from both sides: 3x + 4 = 19.
2. Subtract 4 from both sides: 3x = 15.
3. Divide both sides by 3: x = 5.

Check both sides with x = 5. The left: 5 × 5 = 25, and 25 + 4 = 29. The right: 2 × 5 = 10, and 10 + 19 = 29. Both are 29, so x = 5 is right.

Forming and solving equations

Forming an equation means turning words into algebra, then solving it. The key stage 4 curriculum asks children to turn simple situations into an equation, solve it, and say what the answer means.

Worked example: think of a number

"I think of a number, multiply it by 3 and add 5. The answer is 26." Call the number n. The equation is 3n + 5 = 26. Subtract 5: 3n = 21. Divide by 3: n = 7.

Worked example: a rectangle

A rectangle is 4 cm longer than it is wide, and its perimeter is 28 cm. Call the width x, so the length is x + 4. The perimeter is two widths and two lengths: 2x + 2(x + 4) = 28. Expand and collect: 4x + 8 = 28. Subtract 8: 4x = 20. Divide by 4: x = 5. The rectangle is 5 cm wide and 9 cm long. Check: 2 × 5 = 10, 2 × 9 = 18, and 10 + 18 = 28.

A bar model is a good way to picture problems like these before writing the equation.

Common mistakes when solving equations

• Doing something to one side only. Whatever happens to the left must happen to the right.
• Undoing in the wrong order. In 4x + 3 = 23, deal with the + 3 before the 4. Dividing first means dividing every term.
• Sign slips when moving terms. To get rid of −6, add 6 to both sides; don't take it away.
• Dividing the wrong way. 3x = 21 gives x = 7, not x = ³⁄₂₁.
• Not checking. Putting the answer back in takes a moment and shows straight away whether it's right.

How to help at home

• Play "think of a number". Take turns making up puzzles like "I double my number and take away 7, and get 15." Then write them as equations together.
• Ask for one step per line. Neat working, with one change at a time, is where most marks are won.
• Always check. Ask your child to put their answer back into the question and show both sides match.
• Talk about the balance. "What did you do to that side? So what do you do to this side?"

Solve the equation: questions to try

1. x − 4 = 9
2. 6x = 42
3. 5x − 3 = 32
4. 2(x + 6) = 22
5. 7x − 2 = 3x + 18
6. I think of a number, double it and take away 7. The answer is 15. Form an equation and solve it.

Answers

1. x = 13 (add 4 to both sides)
2. x = 7 (divide both sides by 6)
3. x = 7 (add 3 to get 5x = 35, then divide by 5)
4. x = 5 (divide by 2 to get x + 6 = 11, then subtract 6)
5. x = 5 (subtract 3x to get 4x − 2 = 18, add 2 to get 4x = 20, then divide by 4)
6. 2n − 7 = 15. Add 7: 2n = 22. Divide by 2: n = 11.

Solving equations leads on to inequalities, which are solved in almost the same way. For the rest of what children learn in Years 7 to 9, see the KS3 maths guide.