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Year 10
Higher

Solving a quadratic and linear pair of simultaneous equations using elimination

I can solve two (one linear, one quadratic) simultaneous equations algebraically using elimination.

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Year 10
Higher

Solving a quadratic and linear pair of simultaneous equations using elimination

I can solve two (one linear, one quadratic) simultaneous equations algebraically using elimination.

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Lesson details

Key learning points

  1. When one equation is quadratic it will only be possible to eliminate the linear variable.
  2. Doing so produces a third equation which is quadratic.
  3. You can use any of your methods for solving a quadratic to find possible solutions.
  4. These values need to be substituted into one of the original equations to find its pair.
  5. If your quadratic has two valid solutions then you will have two pairs of solutions to your simultaneous equations.

Keywords

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

Common misconception

Pupils find the two $$x$$ values after solving the combined quadratic and declare that the solution.

It is important to substitute both $$x$$ values back in to one of the original equations to find the corresponding $$y$$ values. Without these, the solutions are incomplete.


To help you plan your year 10 maths lesson on: Solving a quadratic and linear pair of simultaneous equations using elimination, download all teaching resources for free and adapt to suit your pupils' needs...

When doing examples beyond this lesson be sure to show pupils the graphical results of the pairs of equations (one linear, one quadratic) that they solve so that they can see why each $$x$$ value needs a corresponding $$y$$ value.
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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.
If A and B are simultaneous equations such that A: $$2x + 4y = 42$$ and B: $$3x + y = 43$$ where will coordinate pair $$(1, 10)$$ go in the Venn diagram?
An image in a quiz
In B but not A
Correct answer: In A but not B
The intersection A and B
Outside of both A and B
Q2.
If A and B are simultaneous equations such that A: $$4x + 4y = 20$$ and B: $$6x + 2y = 14$$ where will coordinate pair $$(1, 4)$$ go in the Venn diagram?
An image in a quiz
Correct answer: The intersection A and B
In B but not A
In A but not B
Outside of both A and B
Q3.
If A and B are simultaneous equations such that A: $$2y - 2x = 8$$ and B: $$9x + 3y = 12$$ where will coordinate pair $$(0, 0)$$ go in the Venn diagram?
An image in a quiz
In A but not B
In B but not A
The intersection A and B
Correct answer: Outside of both A and B
Q4.
If A and B are simultaneous equations such that A: $$3x + y = 40$$ and B: $$14x + 2y = 176$$ where will coordinate pair $$(12, 4)$$ go in the Venn diagram?
An image in a quiz
In A but not B
In B but not A
Correct answer: The intersection A and B
Outside of both A and B
Q5.
If A and B are simultaneous equations such that A: $$6x + 2y = 26$$ and B: $$7x + 6y = 34$$ where will coordinate pair $$(5, -2)$$ go in the Venn diagram?
An image in a quiz
Correct answer: In A but not B
In B but not A
The intersection A and B
Outside of both A and B
Q6.
If A and B are simultaneous equations such that A: $$6x = 18 - 2y$$ and B: $$8x +6y = 14$$ where will coordinate pair $$(1, 1)$$ go in the Venn diagram?
An image in a quiz
In A but not B
Correct answer: In B but not A
The intersection A and B
Outside of both A and B

Exit quiz

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6 Questions

Q1.
What is the positive value of $$x$$ for $$x^2 + 3x - 4 = 0$$
Correct Answer: 1, x = 1
Q2.
What is the value of $$x$$ for $$2x^2 + 4x + 2 = 0$$
Correct Answer: -1, x=-1
Q3.
What is the negative value of $$x$$ for $$2x^2 + 4x - 6 = 0$$
Correct Answer: -3, x=-3
Q4.
Write the positive coordinate solution for $$x^2 + 3y = 19$$ and $$3x - 3y = -9$$
Correct Answer: (2,5)
Q5.
Write the positive coordinate solution for $$x^2 + 2y = 11$$ and $$3x - 2y = 7$$
Correct Answer: (3,1)
Q6.
Which of these is not a solution for simultaneous equations $$x^2 + 4y = 17$$ and $$2x - 4y = -2$$?
$$(-5, -2)$$
Correct answer: $$(5, 3)$$
$$(3, 2)$$