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Maths explained

Solving linear equations step by step, a simple guide

Learn to solve linear equations step by step: one-step and two-step equations, unknowns on both sides, brackets and fractions, with fully worked examples.

For students. By the Qlive team. . 5 min read.

This guide is for you if you are in roughly Grades 6 to 9 and want a clear method for solving equations. In CBSE you meet simple equations in Class 7 and linear equations in one variable in Class 8. The same skill sits at the heart of Key Stage 3, GCSE and IGCSE algebra, and you will use it in physics and chemistry too. If you can solve linear equations confidently, a lot of later maths becomes easier.

The idea in plain words

An equation is like a balance. The left side and the right side are equal. Your job is to find the value of the unknown (usually x) that keeps it balanced.

The one rule that makes everything work:

Whatever you do to one side, do exactly the same to the other side.

To get x on its own, you undo what has been done to it, using the opposite operation:

If the equation has… Undo it by…
+ 5 subtracting 5 from both sides
- 5 adding 5 to both sides
× 3 (written 3x) dividing both sides by 3
÷ 4 (written x/4) multiplying both sides by 4

Undo additions and subtractions first, then multiplications and divisions. It is the reverse of the order in which the equation was built.

Worked examples

Example 1: a two-step equation

Solve 3x + 5 = 20.

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

Check: 3 × 5 + 5 = 15 + 5 = 20. Correct.

Example 2: x on both sides

Solve 5x - 7 = 2x + 11.

  1. Subtract 2x from both sides to collect the x terms on the left: 3x - 7 = 11
  2. Add 7 to both sides: 3x = 18
  3. Divide both sides by 3: x = 6

Check:

  • Left side: 5 × 6 - 7 = 30 - 7 = 23
  • Right side: 2 × 6 + 11 = 12 + 11 = 23

Both sides are 23, so x = 6 is correct.

Example 3: brackets

Solve 4(x - 3) = 2(x + 5).

  1. Expand the brackets. Multiply every term inside by the number outside: 4x - 12 = 2x + 10
  2. Subtract 2x from both sides: 2x - 12 = 10
  3. Add 12 to both sides: 2x = 22
  4. Divide by 2: x = 11

Check:

  • Left side: 4(11 - 3) = 4 × 8 = 32
  • Right side: 2(11 + 5) = 2 × 16 = 32

Correct.

Example 4: fractions

Solve (x + 2)/3 = (x - 1)/2.

Fractions are easier to clear away first. The denominators are 3 and 2, and their lowest common multiple is 6.

  1. Multiply both sides by 6: 6 × (x + 2)/3 = 6 × (x - 1)/2
  2. Simplify: 2(x + 2) = 3(x - 1)
  3. Expand: 2x + 4 = 3x - 3
  4. Subtract 2x from both sides: 4 = x - 3
  5. Add 3 to both sides: 7 = x, so x = 7

Check:

  • Left side: (7 + 2)/3 = 9/3 = 3
  • Right side: (7 - 1)/2 = 6/2 = 3

Correct.

A method that works every time

For any linear equation, follow these steps in order:

  1. Clear fractions by multiplying every term on both sides by the lowest common multiple of the denominators.
  2. Expand brackets.
  3. Collect x terms on one side and plain numbers on the other.
  4. Divide by the number in front of x.
  5. Check by substituting back into the original equation.

Not every equation needs every step, but the order never changes.

Common mistakes and how to avoid them

Doing something to only one side. If you subtract 5 on the left, you must subtract 5 on the right. Write each step as a full new line, both sides, so nothing gets lost.

Sign errors when moving terms. “Moving” a term across the equals sign is really subtracting or adding it on both sides. When 2x moves from the right to the left, it becomes - 2x. If you say to yourself “subtract 2x from both sides” instead of “move it over”, you will make fewer slips.

Expanding brackets only partly. 4(x - 3) is 4x - 12, not 4x - 3. The number outside multiplies every term inside.

Negative times negative. -2(x - 4) = -2x + 8, because -2 × -4 = +8. This is one of the most common slips in exams.

Clearing fractions on only some terms. In x/4 + 3 = 7, if you multiply by 4, every term gets multiplied: x + 12 = 28. Students often write x + 3 = 28, forgetting to multiply the 3.

Dividing the wrong way. If 3x = 18, then x = 18 ÷ 3 = 6. It is not 3 ÷ 18. The number you divide by is the one in front of x.

Skipping the check. Checking takes seconds and is the easiest way to pick up marks you would otherwise lose.

Try these

  1. Solve 7x - 4 = 45.
  2. Solve 6x + 3 = 2x + 27.
  3. Solve 3(2x - 1) = 4x + 7.
  4. Solve x/4 + 3 = 7.
  5. Solve (2x - 1)/5 = (x + 1)/3.

Answers

  1. Add 4: 7x = 49. Divide by 7: x = 7. Check: 49 - 4 = 45.
  2. Subtract 2x: 4x + 3 = 27. Subtract 3: 4x = 24. Divide by 4: x = 6. Check: 39 on both sides.
  3. Expand: 6x - 3 = 4x + 7. Subtract 4x: 2x - 3 = 7. Add 3: 2x = 10. x = 5. Check: 3 × 9 = 27 and 20 + 7 = 27.
  4. Subtract 3: x/4 = 4. Multiply by 4: x = 16. Check: 16/4 + 3 = 4 + 3 = 7.
  5. Multiply both sides by 15: 3(2x - 1) = 5(x + 1). Expand: 6x - 3 = 5x + 5. Subtract 5x: x - 3 = 5. x = 8. Check: 15/5 = 3 and 9/3 = 3.

Still unsure?

If equations still feel shaky, the problem is often one step underneath, such as negative numbers or fractions. A Qlive teacher can work through it with you one-to-one, starting with a free assessment that finds the exact gap, so your classes go straight to what you need. You can read more about how we teach on our maths page. Once these feel easy, move on to simultaneous equations.

Common questions

What makes an equation linear?

In a linear equation the unknown, usually x, appears only to the power 1. There is no x², no x³ and no x in the bottom of a fraction. If you drew it as a graph, it would be a straight line.

Do I have to move x to the left side?

No. You can collect the x terms on whichever side you like. Many students find it easier to collect them on the side where there are more x's, so the number in front stays positive.

How can I be sure my answer is right?

Put your answer back into the original equation and work out both sides separately. If both sides give the same number, your answer is correct. It takes a few seconds and catches most mistakes.

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