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Year 10
Higher

Solving a quadratic and linear pair of simultaneous equations using substitution

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

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New
New
Year 10
Higher

Solving a quadratic and linear pair of simultaneous equations using substitution

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

Download all resources
Share activities with pupils

Lesson details

Key learning points

  1. Consider which variable to make the subject for the substitution.
  2. Depending which variable you chose, you may need to expand a pair of bracketed expressions.
  3. Your choice may make the substitution significantly easier.

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

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

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.
Which of these is not a solution for simultaneous equations $$2x^2 + 8y = 34$$ and $$x - 2y = -1$$?
$$(-5, -2)$$
$$(3, 2)$$
Correct answer: $$(5, 3)$$
Q2.
Write the positive coordinate solution for $$2x^2 + 6y = 38$$ and $$x - y = -3$$
Correct Answer: (2,5)
Q3.
What is the negative value of $$x$$ for $$4x^2 + 8x - 12 = 0$$
Correct Answer: -3, x = -3
Q4.
What is the value of $$x$$ for $$4x^2 + 8x + 4 = 0$$
Correct Answer: - 1, x = -1
Q5.
What is the negative value of $$x$$ for $$6x^2 + 12x - 18 = 0$$
Correct Answer: -3, x = -3
Q6.
What is the positive value of $$x$$ for $$2x^2 + 6x - 8 = 0$$
Correct Answer: 1, x = 1

Exit quiz

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6 Questions

Q1.
If equation A is $$x^2 + 3y = 24 $$ and equation B is $$4x - 3y = -3$$, which coordinate pair would sit in A and not B?
An image in a quiz
$$(3, 5)$$
Correct answer: $$(6, -4)$$
$$(0, 6)$$
Q2.
If equation A is $$x - 5y = -7$$ and equation B is $$x^2 + y^2 = 13$$, which coordinate pair would sit in A and not B?
An image in a quiz
Correct answer: $$(-2, 1)$$
$$(3, 2)$$
$$(0, 4)$$
Q3.
If equation A is $$x^2 + 4y = 68$$ and equation B is $$x - 4y = 4$$, which coordinate pair would sit in the intersection of A and B?
An image in a quiz
$$(2, 16)$$
$$(6, 8)$$
Correct answer: $$(8, 1)$$
Q4.
What is the positive coordinate pair that solves both $$x^2 + y^2 = 17$$ and $$x - y = 3$$?
Correct Answer: (4,1)
Q5.
What is the positive coordinate pair that solves both $$x^2 + y^2 = 40$$ and $$x - y = 4$$?
Correct Answer: (6,2)
Q6.
What is the positive coordinate pair that solves both $$x^2 + y^2 = 13$$ and $$x - y = -1$$?
Correct Answer: (2,3)