5.1 Solving Equations 1

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: one-step and two-step equations, inverse operations and checking your answer. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Understand that an equation is a balance: both sides have the same value
• Solve one-step equations using inverse operations
• Solve two-step equations, including ones with negative or decimal answers
• Check a solution by substituting it back into the equation

Key words

• Equation: A statement with an equals sign and an unknown, such as 3x + 4 = 19. It is only true for certain values.
• Unknown: The letter whose value you want to find.
• Solve: Find the value of the unknown that makes the equation true.
• Inverse operation: The operation that undoes another: + and − are inverses, × and ÷ are inverses.
• Solution: The value that makes the equation true, e.g. x = 5.

Equations are balances

An equation is like a pair of weighing scales that balance. Whatever you do to one side, you must do to the other side so it stays balanced.

Key fact: Do the same thing to both sides. Use the inverse operation to get the letter on its own.

One-step equations

• x + 8 = 21; Inverse operation subtract 8 from both sides; Solution x = 13
• x − 6 = 14; Inverse operation add 6 to both sides; Solution x = 20
• 7x = 42; Inverse operation divide both sides by 7; Solution x = 6
• x/5 = 9; Inverse operation multiply both sides by 5; Solution x = 45

7x means 7 × x, and x/5 means x ÷ 5. Answers do not have to be whole numbers.

Worked example: a decimal answer

Solve 4x = 26.
Divide both sides by 4: x = 26 ÷ 4 = 6.5

Answer: x = 6.5

Two-step equations

In 3x + 4 = 19, the letter has been multiplied by 3 and then had 4 added. Undo these in reverse order: first subtract 4, then divide by 3.

Worked example: subtract, then divide

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

Answer: x = 5

Worked example: a negative answer

Solve 2y + 13 = 4.
Subtract 13 from both sides: 2y = −9
Divide both sides by 2: y = −4.5

Answer: y = −4.5

Worked example: a division in the equation

Solve m/4 − 3 = 5.
Add 3 to both sides: m/4 = 8
Multiply both sides by 4: m = 32

Answer: m = 32

Checking and harder layouts

Always check by putting your answer back into the original equation. If both sides match, you are right.

Worked example: check a solution

Is x = 5 the solution of 3x + 4 = 19?
Left side: 3 × 5 + 4 = 15 + 4 = 19. Right side: 19. They match.

Answer: yes, x = 5 is correct

Worked example: the letter on the right, and a whole side divided

(a) Solve 31 = 5k + 6. Subtract 6: 25 = 5k. Divide by 5: k = 5
(b) Solve (x + 7)/3 = 6. Multiply by 3: x + 7 = 18. Subtract 7: x = 11

Answer: (a) k = 5 (b) x = 11

Key fact: In (x + 7)/3 the whole of x + 7 is divided by 3, so multiply by 3 first.

Common mistakes

• Doing the wrong inverse: in 3x + 6 = 27 you subtract 6 (3x = 21), you do not add it.
• Undoing in the wrong order: deal with the + or − first, then the × or ÷.
• Only changing one side. Whatever you do, do it to both sides.
• Thinking an answer must be a whole number or positive: x = −4.5 and x = 2.25 are fine.
• Writing 26 ÷ 4 as 4 ÷ 26. In 4x = 26, x = 26 ÷ 4.

Revision questions

Try these, then check your answers below each one.

Question 1. Solve x + 13 = 31.

Answer: 18

Working: x = 31 - 13 = 18

Question 2. Solve 4x = 52.

Answer: 13

Working: x = 52 ÷ 4 = 13

Question 3. Solve 3x - 8 = 19.

Answer: 9

Working: 3x = 27, so x = 9.

Related lessons

• Previous lesson: 4.8 Calculating Percentages 2
• Next lesson: 5.2 Solving Equations 2
• In this chapter: 5.3 Solving Equations with Brackets
• In this chapter: 5.4 Introducing Inequalities
• In this chapter: 5.5 More Inequalities

Continue in the full course

This lesson is section 5.1 (EGCMF0501) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.