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Year 10
Higher

Solving a quadratic and linear pair of simultaneous equations graphically

I can relate the solutions (0, 1 or 2) of a linear and quadratic pair of simultaneous equations to the graphs of these equations.

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New
New
Year 10
Higher

Solving a quadratic and linear pair of simultaneous equations graphically

I can relate the solutions (0, 1 or 2) of a linear and quadratic pair of simultaneous equations to the graphs of these equations.

Download all resources
Share activities with pupils

Lesson details

Key learning points

  1. You can use your knowledge of graphical representations to draw the graphs on the same axes.
  2. If there exists two solutions, then the points of intersection are those solutions.
  3. If there exists one solution, then the point where the linear graph touches the quadratic graph is that solution.
  4. If there are no solutions then the graphs do not touch at all.
  5. When substituting or eliminating, you are left with a combined quadratic which has roots where the graphs intersect.

Common misconception

For a pair of simultaneous equations (one linear and one quadratic), there will always be two intersections and therefore two solutions.

The linear graph might be tangent to the curve therefore only one intersection (solution). Alternatively, the straight line may never intersect the parabola and in this case, there is no solution.

Keywords

  • Linear - The relationship between two variables is linear if, when plotted on a pair of axes, a straight line is formed.

  • Quadratic - A quadratic is an equation, graph, or sequence whereby the highest exponent of the variable is 2; the graph of which forms a parabola.

If you have access to technology this a good opportunity to improve fluency. Get pupils to solve pairs of simultaneous equations (where one is linear and one is quadratic) using a written algebraic method, then use graphing technology to demonstrate or check that their working is correct.
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.
What is the positive coordinate pair that solves both $$x^2 + y^2 = 13$$ and $$9x -9y = 9$$?
Correct Answer: (3,2)
Q2.
What is the positive coordinate pair that solves both $$x^2 + y^2 = 17$$ and $$5x - 5y = 15$$?
Correct Answer: (4,1)
Q3.
What is the positive coordinate pair that solves both $$x^2 + y^2 = 40$$ and $$3x - 3y = 12$$?
Correct Answer: (6,2)
Q4.
If equation A is $$x^2 + 4y = 68$$ and equation B is $$2x - 8y = 8$$, which coordinate pair would sit in the intersection of A and B?
An image in a quiz
$$(2, 16)$$
$$(6, 8)$$
Correct answer: $$(8, 1)$$
Q5.
If equation A is $$2x - 10y = -14$$ 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: $$(-2, 1)$$
$$(3, 2)$$
$$(0, 4)$$
Q6.
If equation A is $$x^2 + 3y = 24 $$ and equation B is $$8x - 6y = -6$$, which coordinate pair would sit in A and not B?
An image in a quiz
$$(3, 5)$$
Correct answer: $$(6, -4)$$
$$(0, 6)$$

Exit quiz

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6 Questions

Q1.
How many solutions satisfy both equations?
An image in a quiz
1
Correct answer: 2
0
Q2.
How many solutions satisfy both equations?
An image in a quiz
Correct answer: 1
2
0
Q3.
How many solutions satisfy both equations?
An image in a quiz
1
2
Correct answer: 0
Q4.
How many solutions satisfy both equations?
An image in a quiz
Correct answer: 1
2
0
Q5.
When $$y = 0$$, how many solutions are there for the equation?
An image in a quiz
1
Correct answer: 2
0
Q6.
When $$y = 0$$, how many solutions are there for the equation?
An image in a quiz
Correct answer: 1
2
0