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Year 8

Many equations, one solution

I can understand that a family of linear equations can all have the same solution.

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New
New
Year 8

Many equations, one solution

I can understand that a family of linear equations can all have the same solution.

Download all resources
Share activities with pupils

Link copied to clipboard

These resources will be removed by end of Summer Term 2025.

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Lesson details

Key learning points

  1. Starting from a single value for a variable you can write equations that are equivalent.
  2. A solution to a linear equation is a value that makes the two sides equal .
  3. All the equivalent equations will have the same solution.

Keywords

  • Solution - A solution to an equality with one variable is a value which, when substituted, maintains the equality between the expressions.

  • Equation - An equation is used to show two expressions that are equal to each other.

Common misconception

Pupils may think a value has to be added to each term in an expression.

When adding a term to an expression we can write it as a single addition and then collect like terms if necessary. Show with numbers first


To help you plan your year 8 maths lesson on: Many equations, one solution, download all teaching resources for free and adapt to suit your pupils' needs...

Give pupils a simple equation e.g. x = 9 and see how many equivalent equations they can generate in a set time. For challenge you could ask them to generate ones with 2 steps, 3 steps, a division, a fraction etc.
Teacher tip

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Starter quiz

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6 Questions

Q1.
This bar model represents the initial equation $$2x + 3 = 9$$. What needs to be done to the top bar to maintain equality?
An image in a quiz
Correct answer: $$+9$$
$$+2x$$
Correct answer: $$+(2x+3)$$
$$\times 1$$
Correct answer: $$\times 2$$
Q2.
Which of these calculations show '6 plus 3 all multiplied by 2'?
6 + 3 × 2
Correct answer: 2(6 + 3)
2 × 6 + 3
Correct answer: 2 × 6 + 2 × 3
6 + 3 + 6
Q3.
Lucas manipulates the equation $$2x + 8 = 6$$. Which of these equations could he obtain and maintain equality?
Correct answer: $$x + 4 = 3$$
Correct answer: $$2x + 10 = 8$$
$$10x + 8 = 30$$
$$2x = 6$$
Correct answer: $$4x + 8 = 2x + 6$$
Q4.
Which expression could fill the blank on the right hand side of this equation to maintain equality?
An image in a quiz
Correct answer: $$2x + 1$$
$$2x + 2$$
$$x + 4$$
Correct answer: $${2x + 4}\over 2$$
Correct answer: $$x + 2$$
Q5.
Which of these equations has a solution of $$x = 6$$?
$$x + 5 = −1$$
Correct answer: $$2x − 3 = 9$$
Correct answer: $$5 + 3x = 2x + 11$$
$$10 − x = 16$$
$$15 − 2x = 2x$$
Q6.
Which of these is a solution to the equation $$18 − 6x = 12 − 4x$$?
$$x = −4$$
$$x = −1$$
$$x = 0$$
$$x = 2$$
Correct answer: $$x = 3$$

Exit quiz

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6 Questions

Q1.
Which of these equations are equivalent to $$x = 5$$?
Correct answer: $$2x = 10$$
$$x − 2 = 7$$
Correct answer: $$x + 3 = 8$$
Correct answer: $$2x = x + 5$$
$$x + 5 = 0$$
Q2.
If $$8x + 4 = 12$$, which of these equations has maintained equality?
$$2x + 4 = 3$$
$$2x + 1 = 12$$
Correct answer: $${1\over 4}(8x + 4) = {12\over 4}$$
$$8x + 1 = 3$$
Correct answer: $${{8x + 4}\over 4} = 3 $$
Q3.
Which of these equations are equivalent to $$12x + 2 = 8x − 6$$?
Correct answer: $$12x + 8 = 8x$$
$$12x = 8x − 4$$
Correct answer: $$6x + 1 = 4x − 3$$
Correct answer: $$10x + 2 = 6x − 6$$
$$3x + 1 = 2x + 3$$
Q4.
Which of these is a solution to the equation $$5x − 5 = 9 − 2x$$?
$$x = −2$$
$$x = −1$$
$$x = 0$$
$$x = 1$$
Correct answer: $$x = 2$$
Q5.
What operation is missing from this diagram?
An image in a quiz
$$\times 4$$
Correct answer: $$\times {1\over 4}$$
Correct answer: $$\div 4$$
$$ + (−9)$$
Q6.
What operation is missing from this diagram?
An image in a quiz
$$\times 2$$
$$+ (−3)$$
$$+ x$$
$$+ (x + 3)$$
Correct answer: $$+ (x − 3)$$