5.2 Solving Equations 2

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: equations with the unknown on both sides, and writing equations from words and shapes. 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

• Solve equations with the unknown on both sides
• Deal with negative terms in the unknown, such as 7 − 2x
• Write an equation from a description in words and solve it
• Write and solve equations from shapes, then use the answer

Key words

• Term: A number, a letter, or numbers and letters multiplied, e.g. 5x, −2x, 7.
• Like terms: Terms with the same letter part, e.g. 5x and 2x. They can be collected.
• Form an equation: Write the information in a problem as an equation using a letter for the unknown.
• Coefficient: The number in front of the letter. In 5x the coefficient is 5.

The unknown on both sides

When the letter appears on both sides, first get all the letter terms on one side. Take away the smaller letter term from both sides, so you keep a positive number of xs. Then solve as a two-step equation.

Worked example: collect the x terms on one side

Solve 5x + 3 = 2x + 18.
Subtract 2x from both sides: 3x + 3 = 18
Subtract 3 from both sides: 3x = 15
Divide both sides by 3: x = 5
Check: 5 × 5 + 3 = 28 and 2 × 5 + 18 = 28 ✓

Answer: x = 5

Worked example: a negative answer

Solve 8y + 5 = 3y − 10.
Subtract 3y: 5y + 5 = −10. Subtract 5: 5y = −15. Divide by 5: y = −3

Answer: y = −3

Negative letter terms

If one side has −2x, add 2x to both sides. This removes the negative and keeps the x terms positive.

Worked example: add the negative term to both sides

Solve 7 − 2x = 3x − 8.
Add 2x to both sides: 7 = 5x − 8
Add 8 to both sides: 15 = 5x
Divide by 5: x = 3

Answer: x = 3

Worked example: subtract the smaller x term

Solve 4x − 9 = 6x − 21.
4x is smaller, so subtract 4x: −9 = 2x − 21
Add 21: 12 = 2x. Divide by 2: x = 6

Answer: x = 6

Key fact: It does not matter which side the letter ends up on: 12 = 2x means the same as 2x = 12.

Forming equations from words

Choose a letter for the unknown. Write each description as an expression, then put the two equal expressions either side of =.

Worked example: think of a number

I think of a number. Doubling it and adding 11 gives the same answer as trebling it and subtracting 4.
Let the number be n. 2n + 11 = 3n − 4
Subtract 2n: 11 = n − 4. Add 4: n = 15

Answer: the number is 15

Worked example: comparing two prices

Gym A charges £20 to join plus £15 a month. Gym B charges £50 to join plus £9 a month. After how many months do they cost the same?
Let m be the number of months. 20 + 15m = 50 + 9m
Subtract 9m: 20 + 6m = 50. Subtract 20: 6m = 30. m = 5
Check: 20 + 75 = £95 and 50 + 45 = £95 ✓

Answer: after 5 months (both cost £95)

Forming equations from shapes

Shapes give you facts you can turn into equations. Opposite sides of a rectangle are equal, so their expressions are equal.

Worked example: opposite sides are equal

5x − 6 = 3x + 2
Subtract 3x: 2x − 6 = 2. Add 6: 2x = 8. x = 4
Length = 3 × 4 + 2 = 14 cm. Width = 2 × 4 = 8 cm.
Perimeter = 14 + 8 + 14 + 8 = 44 cm

Answer: x = 4, perimeter 44 cm

Key fact: Finding x is often not the end. Read the question again: you may need to use x to find a length, a cost or an age.

Common mistakes

• Moving a term without changing the operation: 8x − 3 = 5x + 9 gives 3x = 9 + 3, not 9 − 3.
• Subtracting the larger letter term and then losing track of the negative sign.
• Writing x = 15/5 as the final answer. Simplify it: x = 3.
• Stopping at x when the question asks for a length, a price or a perimeter.

Revision questions

Try these, then check your answers below each one.

Question 1. Solve 6x + 4 = 2x + 24.

Answer: 5

Working: 4x + 4 = 24, 4x = 20, x = 5.

Question 2. Solve 9y - 7 = 4y + 13.

Answer: 4

Working: 5y - 7 = 13, 5y = 20, y = 4.

Question 3. Solve 15 - 2x = x + 3. (Choose from: x = 4, x = 6, x = -4, x = 12.)

Answer: x = 4

Working: 15 = 3x + 3, 12 = 3x, x = 4.

Related lessons

• Previous lesson: 5.1 Solving Equations 1
• Next lesson: 5.3 Solving Equations with Brackets
• In this chapter: 5.4 Introducing Inequalities
• In this chapter: 5.5 More Inequalities
• In this chapter: 5.6 Using Formulae

Continue in the full course

This lesson is section 5.2 (EGCMF0502) 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.