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Linear equations practice

Solve a linear equation by making equivalent changes to both sides until the variable is isolated. Distribution and negative signs are often the real challenge.

By ExamCat · Original study lesson · Published September 24, 2026

Understand the method

Adding or subtracting the same quantity preserves equality. Multiplying or dividing both sides by the same nonzero number does too. Before undoing operations, simplify each side carefully; a minus sign outside parentheses affects every term inside.

Worked example, step by step

Solve 3(2x − 4) + 5 = 17.

  1. Distribute three to both terms: 6x − 12 + 5 = 17.
  2. Combine constants: 6x − 7 = 17.
  3. Add seven, then divide by six: 6x = 24, so x = 4.
  4. Check in the original: 3(8 − 4) + 5 = 12 + 5 = 17.

A mistake to watch for

Checking only a later line can hide a mistake made while distributing. Substitute into the original equation. If variables disappear, decide whether the remaining statement is always true or impossible.

Try it yourself: three questions

Write an answer before opening the explanation. For calculations, keep your units and intermediate steps; for concepts, state the reason for your choice.

1. Solve 5x − 9 = 16.

Show answer and explanation

Add nine to get 5x = 25; divide by five to get x = 5.

2. Solve 2(x + 3) = 2x + 6.

Show answer and explanation

Both sides simplify to the same expression. Every real x works, so there are infinitely many solutions.

3. Solve 2x + 1 = 2x + 4.

Show answer and explanation

Subtracting 2x gives 1 = 4, which is impossible. There is no solution.

Build your next practice session

Mark the first step where your reasoning differed from the explanation. Turn that step into a question, answer it without looking, then change one condition in the example and explain what would change. A correct guess is a reason to revisit the method.

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