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      Many equations, one solution

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

      Lesson 6 of 17
      New
      New
      • Year 8

      Many equations, one solution

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

      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

      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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      Prior knowledge starter quiz

      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: $$x = 3$$

      Assessment exit quiz

      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: $$+ (x − 3)$$

      Lesson appears in

      UnitMaths / Solving linear equations