Solving a quadratic and linear pair of simultaneous equations graphically | Maths

Starter quiz

What is the positive coordinate pair that solves both $x^2 + y^2 = 13$ and $9x -9y = 9$ ?

'(3,2)' ✓
What is the positive coordinate pair that solves both $x^2 + y^2 = 17$ and $5x - 5y = 15$ ?

'(4,1)' ✓
What is the positive coordinate pair that solves both $x^2 + y^2 = 40$ and $3x - 3y = 12$ ?

'(6,2)' ✓
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?

$(2, 16)$

$(6, 8)$

$(8, 1)$ ✓
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?

$(-2, 1)$ ✓

$(3, 2)$

$(0, 4)$
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?

$(3, 5)$

$(6, -4)$ ✓

$(0, 6)$

Exit quiz

How many solutions satisfy both equations?

1

2 ✓

0
How many solutions satisfy both equations?

1 ✓

2

0
How many solutions satisfy both equations?

1

2

0 ✓
How many solutions satisfy both equations?

1 ✓

2

0
When $y = 0$ , how many solutions are there for the equation?

1

2 ✓

0
When $y = 0$ , how many solutions are there for the equation?

1 ✓

2

0

Worksheet

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Lesson Details

Key learning points

You can use your knowledge of graphical representations to draw the graphs on the same axes.
If there exists two solutions, then the points of intersection are those solutions.
If there exists one solution, then the point where the linear graph touches the quadratic graph is that solution.
If there are no solutions then the graphs do not touch at all.
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.