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Linear equations: worked examples and online practice

Basic linear equations
2x = 8
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Basic linear equations

  1. 01 2x = 8 Start practising
  2. 02 7x + 3 = 2x + 13 Start practising
  3. 03 10 − 2x = 4 Start practising
  4. 04 2x − 3(4 − x) = 8 Start practising
  5. 05 4(2x − 3) + 5 = 13 Start practising
  6. 06 3x + 5 = 14 Start practising

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Linear equations with fractions

  1. 01 x2 = 5 Start practising
  2. 02 x2 + x3 = 10 Start practising
  3. 03 2x + 15 = 3 Start practising
  4. 04 2x3 + 12 = 5x6 Start practising
  5. 05 x − 12 + x + 13 = 4 Start practising
  6. 06 3x − 24 = 2x + 13 Start practising

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Equations with the unknown in the denominator

  1. 01 10x = 2 Start practising
  2. 02 x + 2x = 3 Start practising
  3. 03 2x − 3 = 4x + 1 Start practising
  4. 04 x + 1x − 1 = 2 Start practising
  5. 05 52x + 1x = 72 Start practising
  6. 06 1x + 12x = 34 Start practising

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Systems of linear equations

  1. 01 x + y = 10x − y = 2 Start practising
  2. 02 2x + y = 7x − y = 2 Start practising
  3. 03 x + 2y = 83x − 2y = 4 Start practising
  4. 04 3x + y = 9x + 2y = 8 Start practising
  5. 05 2x − 3y = 1x + y = 3 Start practising
  6. 06 4x + 2y = 14x − y = 1 Start practising

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Word problems with linear equations

  1. 01 I'm thinking of a number. If I add 10 to it, I get 25. Start practising
  2. 02 Twice a number, increased by 5, is 17. Start practising
  3. 03 The sum of three consecutive numbers is 33. Start practising
  4. 04 The father is 3 times older than the son. Together they are 48 years old. Start practising
  5. 05 The length of a rectangle is 4 cm greater than its width. The perimeter is 40 cm. Start practising
  6. 06 There are 30 pupils in the class. There are 4 more girls than boys. Start practising

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Equations with absolute value

  1. 01 |x| = 5 Start practising
  2. 02 |x + 1| = 4 Start practising
  3. 03 |x − 2| = 3 Start practising
  4. 04 |2x + 3| = 7 Start practising
  5. 05 |3x − 1| = 5 Start practising
  6. 06 |x − 2| = −1 Start practising

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Equations with a parameter

  1. 01 px = 5 Start practising
  2. 02 2x − px = 6 Start practising
  3. 03 x(p + 2) = p + 2 Start practising
  4. 04 px − 4 = 2x Start practising
  5. 05 px + x = 4 Start practising
  6. 06 p(x + 2) = 5(x + 2) Start practising

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Nové!

Linear equations from entrance exams

  1. 01 2 − x2 + 2x = 2,5x − 3 Start practising
  2. 02 4 − 7 − 3y5 = 3 + 7y − 410 Start practising
  3. 03 y + 5y6 = 2y − 14 + y + 12 Start practising
  4. 04 0,1x + 5(0,04x − 3,2) = 4 − 0,7x Start practising
  5. 05 5x + 215 + x15 = 23x − 35 Start practising
  6. 06 0,3(2x + 1) = 0,2x − 0,7 Start practising

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For teachers: Linear equations worksheet to download as PDF

Download examples on linear equations as a PDF.

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How to solve linear equations — a five-step method

  1. 01

    Remove the fractions

    Short description of the step

  2. 02

    Multiply out the brackets

    Short description of the step

  3. 03

    Move the terms

    Short description of the step

  4. 04

    Divide

    Short description of the step

  5. 05

    Check by substitution

    Short description of the step

The most common mistakes when solving linear equations

  1. 1

    Signs when multiplying out a bracket with a negative number

    What it looks like. In the equation 2x − 3(4 − x) = 8 the bracket gets multiplied out as −12 − 3x, even though the correct result is −12 + 3x. The sign in front of the bracket applies to every term inside it, including the one that already has a minus.

    How to avoid it. Multiply each term in the bracket separately and write down the resulting sign straight away: −3 · 4 = −12 and −3 · (−x) = +3x. That gives 2x − 12 + 3x = 8, so x = 4.

  2. 2

    Multiplying only one side of the equation, or forgetting a term without a fraction

    What it looks like. In the equation x2 + 3 = 5 you multiply the fraction by two but leave the 3 alone: x + 3 = 10. You get x = 7, but the check doesn't confirm it: 72 + 3 = 6.5, not 5.

    How to avoid it. Multiply the whole equation, meaning every term on both sides: 2 · (x2 + 3) = 2 · 5, which gives x + 6 = 10 and x = 4. It helps to put the whole side in brackets before multiplying it.

  3. 3

    Missing condition for the unknown in the denominator

    What it looks like. The equation xx − 2 = 2x − 2 gives x = 2 once you multiply out the denominator. But for x = 2 the denominator is zero, and you can't work with a fraction that has zero on the bottom, so this equation has no solution.

    How to avoid it. Before you start rearranging, write down what x must not equal (here x ≠ 2), and check the result against that condition at the end. If the result breaks the condition, discard it.

  4. 4

    Dividing by an expression that could be zero (with a parameter)

    What it looks like. In the equation x(p + 2) = p + 2 you divide both sides by p + 2 and get x = 1. But for p = −2, p + 2 = 0, and you can't divide by zero: the equation turns into 0 · x = 0, which is true for every x.

    How to avoid it. Before dividing by an expression containing a parameter, ask whether it could be zero. If it could, solve that case separately (here p = −2), and you can divide for the other values of the parameter.

  5. 5

    x calculated in a word problem, but the question left unanswered

    What it looks like. The problem: “The father is 3 times older than the son. Together they are 48 years old. How old is the father?” You call the son’s age x, and the equation x + 3x = 48 gives x = 12. That isn’t the answer yet, though, because 12 years is the son’s age, not the father’s.

    How to avoid it. At the start, write down what x stands for, and at the end go back to the question and answer it in a full sentence: the father is 3 · 12 = 36 years old. Check: 12 + 36 = 48.

Frequently asked questions

What is a linear equation?

A linear equation is an equation where the unknown appears only to the first power — no x², no square root of x, and so on. After rearranging, it can be written as ax + b = 0, where a is a non-zero number. For example, 3x + 5 = 14 is a linear equation and x² = 9 is not.

How do you solve a linear equation?

The procedure has five steps: remove the fractions, multiply out the brackets, move the terms with the unknown to one side and the numbers to the other, divide by the number in front of the unknown, and check the result by substitution. For example, in the equation 3x + 5 = 14 you move the 5 to the right side (3x = 9), divide by three (x = 3) and check: 3 · 3 + 5 = 14.

How do I know I've solved the equation correctly?

By checking it. Substitute the result into the original equation, not the rearranged one, and work out both sides separately. If you get the same number on both sides, the result is correct. In the equation 7x + 3 = 2x + 13 you get x = 2: the left side is 7 · 2 + 3 = 17 and the right side 2 · 2 + 13 = 17.

What if the equation has no solution, or infinitely many?

Sometimes the unknown disappears while you rearrange. If a false statement is left, the equation has no solution: 2x + 1 = 2x + 3 leads to 1 = 3. If a true statement is left, the equation holds for every number: 2(x + 1) = 2x + 2 leads to 0 = 0. |x − 2| = −1 has no solution either, because an absolute value is never negative.

From which grade are linear equations taught?

The basics of linear equations are usually covered in lower secondary school, and the exact grade varies by school curriculum. More advanced types, such as systems of equations or equations with a parameter, come later.

What's the difference between a linear equation and a system of linear equations?

A linear equation has one unknown, and you're looking for one number. A system most often has two unknowns, x and y, and two equations that must hold at the same time, so you're looking for a pair of numbers. The system x + y = 10 and x − y = 2 has the solution x = 6 and y = 4.

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