7x + 12 = 61: take 12 from both sides, 7x = □; divide both by 7, x = □. Fill in each blank

49, 7. 61 − 12 = 49, 49 ÷ 7 = 7. Check with x = 7: 7x + 12 = 49 + 12 = 61.

x = 10 is a solution of 5x + 6 = 51. True or false?

False. Put in x = 10: 5 × 10 + 6 = 56. It is a solution only if that equals 51.

6x + 30 = 26, x = □ Answer in simplest form

−2/3. 6x = −4, so x = −2/3. A solution need not be whole: write it as a fraction in its lowest terms.

Linear equations in one unknown · Ages 12–13

Equations and the balance

Do the same to both sides and it stays level.

How to do it

Method

Do the same to both sides and it stays level

3x + 5 = 20, 3x = 15, x = 5

Another way

Work backwards

20 − 5 = 15, 15 ÷ 3 = 5

In full

An equation is like a level balance: its two sides are equal. The value of x that makes them equal is the solution. The balance method: do the same to both sides and it stays level. 3x + 5 = 20: take 5 from both sides, 3x = 15; divide both by 3, x = 5. Or work backwards: x is multiplied by 3, then 5 is added, to make 20, so go back from 20: take 5, divide by 3, and again 5. Then put the answer back to check: 3 × 5 + 5 = 20. A solution can be negative or a fraction: 3x + 7 = 1, 3x = −6, x = −2.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
7x + 12 = 61: take 12 from both sides, 7x = ; divide both by 7, x = .
Fill in each blank
Show the answer and working
Answer: 49, 7
61 − 12 = 49, 49 ÷ 7 = 7. Check with x = 7: 7x + 12 = 49 + 12 = 61.
Example 2
x = 10 is a solution of 5x + 6 = 51.
True or false?
Show the answer and working
Answer: False
Put in x = 10: 5 × 10 + 6 = 56. It is a solution only if that equals 51.
Example 3
6x + 30 = 26, x =
Answer in simplest form
Show the answer and working
Answer: −23
6x = −4, so x = −23. A solution need not be whole: write it as a fraction in its lowest terms.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.x = 10 is a solution of 6x + 6 = 66.
True or false?
Answer:
2.6x + 29 = −25: take 29 from both sides, 6x = ; divide both by 6, x = .
Fill in each blank
Answer:
3.x5 − 5 = 3, x =
Answer:
4.4 − 6x = 16, x =
Answer:
5.4x + 19 = 14, x =
Answer in simplest form
Answer:

All the answers

1. True
2. −54, −9
3. 40
4. −2
5. −54

Free worksheet: Equations and the balanceWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake