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      Problem solving with simultaneous equations

      Lesson details

      Learning outcome

      I can use my knowledge of simultaneous equations to solve problems.

      Key learning points

      1. Where a problem gives multiple ways of connecting two things, it may mean simultaneous equations can be written.
      2. If simultaneous equations can be written, it is important to state what each variable stands for.
      3. The solutions should always be given in context.

      Keywords

      • Substitution - Substitute means to put in place of another. In algebra, substitution can be used to replace variables with values, terms, or expressions.

      • Elimination - Elimination is a technique to help solve equations simultaneously and is where one of the variables in a problem is removed.

      Common misconception

      Simultaneous equations cannot be applied usefully to real-world scenarios.

      The skill of solving a pair of simultaneous equations can be applied to a wide variety of problems and provides us with answers in a wide variety of contexts.

      Teacher tip

      Pupils see a variety of problems in this lesson. Ask pupils to start with their own answers, work backwards, and come up with their own problems to give to each other to solve.

      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.
      Solve using any method: $$5x + 4y = 11$$ and $$2x + 5y = 35$$. Write your answer in the form $$(x,y)$$.

      Correct Answer: (5, 9)

      Q2.
      Solve using any method: $$x + 5y =24$$ and $$10x + 6y = 20$$. Write your answer in the form $$(x,y)$$.

      Correct Answer: (-1, 5)

      Q3.
      Solve using any method: $$8x + 2y =10$$ and $$6x - 8y = 36$$. Write your answer in the form $$(x,y)$$.

      Correct Answer: (2, -3)

      Q4.
      Solve using any method: $$12x + 4y = 60$$ and $$12x - 10y =102$$. Write your answer in the form $$(x,y)$$.

      Correct Answer: (6, -3)

      Q5.
      Solve using any method: $$3x + y = 15$$ and $$10x - 10y = 90$$. Write your answer in the form $$(x,y)$$.

      Correct Answer: (6, -3)

      Q6.
      Solve using any method: $$3x + 5y = -6$$ and $$10x - 10y = 60$$. Write your answer in the form $$(x,y)$$.

      Correct Answer: (3, -3)

      6 Questions

      Q1.
      Which two equations represent this scenario: Nine small bags (s) and five bigger bags (b) contains 182 counters in total. Two small bags and four bigger bags contain 104 counters in total.

      $$9s + 5b = 104$$
      $$2s + 4b = 182$$
      Correct answer: $$2s + 4b = 104$$
      Correct answer: $$9s + 5b = 182$$

      Q2.
      Which two equations represent this scenario: Five small bags (s) and three bigger bags (b) contains 94 counters in total. Seven small bags and four bigger bags contain 128 counters in total.

      $$5s + 3b = 7s + 4b$$
      Correct answer: $$5s + 3b = 94$$
      $$7s + 3b = 94$$
      Correct answer: $$7s + 4b = 128$$

      Q3.
      A bakery sells cookies in boxes of small and large sizes. Four small and three large boxes contain 235 cookies. Two small boxes and five large boxes contain 275 cookies. How many are in each box?

      Small: 30, Large: 40
      Small: 20, Large: 50
      Correct answer: Small: 25, Large: 45

      Q4.
      A farmer sells bags of apples and oranges. Three bags of apples and four bags of oranges weigh 73 kg. Five bags of apples and three bags of oranges weigh 85 kg. What is the weight of each bag?

      Apples: 9 kg, Oranges: 11 kg
      Correct answer: Apples: 11 kg, Oranges: 10 kg
      Apples: 10 kg, Oranges: 12 kg

      Q5.
      A company produces two types of gadget: standard (s) and premium (p). Selling 5 s and 3 p gadgets makes £260. Selling 3 s and 5 p gadgets makes £220. How much does each type of gadget cost?

      Correct answer: Standard: £40, Premium: £20
      Standard: £50, Premium: £20
      Standard: £40, Premium: £30

      Q6.
      A company produces two types of gadget: standard (s) and premium (p). Selling 6 s and 4 p gadget makes £560. Selling 4 s and 6 p gadgets makes £640. How much does each type of gadget cost?

      Standard: £60, Premium: £80
      Correct answer: Standard: £40, Premium: £80
      Standard: £80, Premium: £100

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