Graphs 7

4 marks · Hard

Consider the simultaneous equations and .

Explain, step by step, how you would use graphs to solve these equations simultaneously, and write down the solution(s) for and .

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

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(x, y) = (1, 0) and (−2, −6)

Must show: Draw both y = x^2 - 4 and y = 2x - 2 on same axes, State that intersections are the solutions, Find both points of intersection, Write down both (x, y) values

Also accept: x = 1, y = 0; x = -2, y = -6; (1, 0), (-2, -6)

Common mistakes: Finding only one solution; Setting y-values equal but not explaining the graphical interpretation

Show mark scheme and worked solution

Marking points

M1 · 1 mark · Describes drawing graphs of both equations on the same set of axes.

M2 · 1 mark · Explains that the intersection points are the solutions.

A1 · 1 mark · Gives one correct pair of solutions for (x, y).

A2 · 1 mark · Gives the second correct pair of solutions for (x, y).

Worked solution

Step 1: Draw the graphs of y = x^2 - 4 (a parabola) and y = 2x - 2 (a straight line) on the same axes.

Step 2: Find the points where the two graphs intersect; these give the solutions to the simultaneous equations.

Step 3: Set x^2 - 4 = 2x - 2, so x^2 - 2x - 2 = 0. Factorising or using the quadratic formula gives x = 1 or x = -2.

Step 4: Substitute back:

If x = 1: y = 2*1 - 2 = 0. If x = -2: y = 2*(-2) - 2 = -4 - 2 = -6.

So the solutions are (1, 0) and (-2, -6).