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

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.

Keywords

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

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.

Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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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?

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?

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?

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?

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?

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?

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: $$(5, 3)$$

$$(3, 2)$$