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      Solution to an equation

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      Lesson details

      Learning outcome

      I can appreciate that a solution is a value that makes the two sides of an equation balance.

      Key learning points

      1. In an expression the variable can take any value.
      2. In a linear equation there is a particular value that makes the equation balance.
      3. The value is the solution to the linear equation.
      4. When graphing a linear expression this value represents a single point.

      Keywords

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

      Common misconception

      When reading solutions from a graph pupils may read the solution as the y-coordinate.

      The solution to the equation is the value of the variable that makes both sides balance.

      Teacher tip

      Using graphing software will allow pupils to explore solutions without the restrictions of drawing the lines themselves. They can write tricky equations for each other to solve and explore when there are no, one or infinite solutions.

      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

      Download starter quiz questions (PDF)

      6 Questions

      Q1.
      What is the value of the expression $$5x − 4$$ when $$x = 3$$?

      Correct Answer: 11, eleven

      Q2.
      What is the value of the expression $$5x − 4$$ when $$x = −1$$?

      Correct Answer: −9, − 9, -9, - 9

      Q3.
      The table of values for the equation $$y = 7 − 2x$$ is shown. The missing number is .

      An image in a quiz
      Correct Answer: 11, eleven, y=11, y = 11

      Q4.
      To find the value of the expression $$21 − 1.5a$$ when $$a = 5$$, you should the variable $$a$$ for the number 5.

      Correct Answer: substitute

      Q5.
      The graph shows 5 lines labelled A, B, C, D and E. Match each line to the equation for that line.

      An image in a quiz
      Correct Answer:A (blue),$$x = −1$$

      $$x = −1$$

      Correct Answer:B (pink),$$y = x$$

      $$y = x$$

      Correct Answer:C (purple),$$y = −1$$

      $$y = −1$$

      Correct Answer:D (green),$$y = 1$$

      $$y = 1$$

      Correct Answer:E (black),$$y = −x$$

      $$y = −x$$

      Q6.
      Match each fraction calculation to its answer.

      Correct Answer:$$4\times {1\over 2}$$,2

      2

      Correct Answer:$$4\times {3\over 2}$$,6

      6

      Correct Answer:$$4\times {5\over 2}$$,10

      10

      Correct Answer:$$4\times {1\over 4}$$,1

      1

      Correct Answer:$$4\times {3\over 4}$$,3

      3

      Correct Answer:$$4\times {5\over4}$$,5

      5

      Assessment exit quiz

      Download exit quiz questions (PDF)

      6 Questions

      Q1.
      Which of these best describes what a solution is?

      The value of the expressions on both sides of the equals sign when the same.
      The value of the expressions of both sides when the variable is zero.
      Correct answer: The value of the variable which makes both sides of the equation equal.
      The value of the variable which makes both sides equal to zero.

      Q2.
      Which of these equations have a solution when $$x = 4$$?

      $$4x = 4$$
      $$x + 4 = 0$$
      Correct answer: $$2x + 1 = 9$$
      $$3x − 15 = 3$$
      Correct answer: $$ 7 − 2x = -1$$

      Q3.
      Which of these is a solution to the equation $$3a + 5 = 2a + 2$$?

      $$a = −5$$
      Correct answer: $$a = −3$$
      $$a = 0$$
      $$a = 2$$
      $$a = 4$$

      Q4.
      Which of these equations have no solution?

      $$2x + 4 = 3x + 4$$
      $$ x + 5 = 10 − x$$
      Correct answer: $$5x + 1 = 5x + 15$$
      $$ 2(x + 4) = 3x + 5$$
      Correct answer: $$4(2x + 1)= 8x$$

      Q5.
      The graph shows the lines with equations $$y = 2 − 2x$$ and $$y = 2x - 2$$. What is the solution to the equation $$2x − 2 = 2 − 2x$$?

      An image in a quiz
      $$x = 2$$
      Correct answer: $$x = 1$$
      $$ x = 0$$
      $$x = −2$$

      Q6.
      Use the graph to match each equation to its solution.

      An image in a quiz
      Correct Answer:$$1 − x = 2$$,solution when $$x = −1$$

      solution when $$x = −1$$

      Correct Answer:$$1 − x = −4$$,solution when $$x = 5$$

      solution when $$x = 5$$

      Correct Answer:$$2x = 2$$,solution when $$x = 1$$

      solution when $$x = 1$$

      Correct Answer:$$2x = −4$$,solution when $$x = −2$$

      solution when $$x = −2$$


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