Linear Equations

Solve one-step and two-step equations with inverse operations, then check each answer by substitution.

About 3 min read
  • Solve one-step equations using inverse operations
  • Solve two-step equations such as 3x + 2 = 14
  • Check an answer by substituting it back
Before you start

You know that a letter can stand for an unknown number, and that adding undoes subtracting and multiplying undoes dividing.

The balance idea

An equation says that two expressions are equal. Think of the equals sign as the middle of a balanced scale. If you change one side, you must change the other side in the same way, or the scale tips. This rule lets us remove everything around the unknown until it stands alone.

Definition: solving an equation

Finding the value of the unknown that makes the equation true.

Definition: inverse operation

The operation that undoes another. Subtraction undoes addition and division undoes multiplication.

One-step equations

In x + 5 = 12, the + 5 is in the way. Undo it by subtracting 5 from both sides:

Maths
x + 5 = 12
x = 12 − 5
x = 7

Two-step equations

3x + 2 = 14 has two operations on x: multiply by 3, then add 2. Undo them in reverse order.

Solve a two-step equation

Solve 3x + 2 = 14.

Maths
3x + 2 = 14
3x = 12        subtract 2 from both sides
x = 4          divide both sides by 3

Check: 3 × 4 + 2 = 14, so the answer is correct.

Always check

Put your answer back into the original equation. It takes seconds and catches most mistakes.

Try it yourself

Type a linear equation below and watch each balancing step.

Linear equation solverType any linear equation in x — it is solved step by step, entirely in your browser.

Press Solve to see each step — then cover the steps and try the next one yourself.

Exam focus
Say it like this“2x − 3 = 11, so 2x = 14, so x = 7”
Not like this“2x − 3 = 11, so x = 7” — Show each step. Method marks can be lost when working is missing.
Common mistakeDoing an operation to one side only, or undoing the operations in the wrong order.

Practice

Solve x + 9 = 20.
Subtract 9 from both sides: x = 11.
Solve 5x = 35.
Divide both sides by 5: x = 7.
Solve 2x − 3 = 11.
Add 3: 2x = 14. Divide by 2: x = 7.
Going further
I think of a number, multiply it by 4 and subtract 6. The answer is 26. Find the number.
4x − 6 = 26, so 4x = 32, so x = 8.

Quick check

Linear equations quick check

What should you do first to solve x + 7 = 15?

Subtraction undoes the + 7, which leaves x alone on one side.

Solve 4x + 1 = 21.

Subtract 1 to get 4x = 20, then divide both sides by 4.

Why do we substitute the answer back into the equation?

If both sides match after substitution, the solution works. If they differ, there is a mistake.
Key takeaways
  • Solving means finding the value that makes the equation true.
  • Do the same thing to both sides.
  • Undo operations in reverse order.
  • Check by substituting your answer.
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