Graphs 2

5 marks · Hard

The lines y = 2x + 3 and y = -x + k intersect at the point (1, 5).

(a)

Find the value of k.

[2 marks]

(b)

Verify that (1, 5) lies on both lines.

[2 marks]

(c)

Determine the equation of the line that passes through the origin and the point of intersection.

[1 mark]

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.

Show answer

(a) k = 6

(b) Verified: y = 2x + 3 gives 5, y = -x + 6 gives 5 for x = 1

Also accept: y = 5x

Common mistakes: Incorrect substitution for k; Incorrect use of points in equation of line

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Correct substitution of x = 1, y = 5 into y = -x + k.

A1 (a) · 1 mark · Correct value for k. (follow-through)

M2 (b) · 1 mark · Correct substitution into y = 2x + 3 and y = -x + k.

A2 (b) · 1 mark · Clearly states (1, 5) lies on both lines with working.

B1 (c) · 1 mark · Finds the equation for the line through (0,0) and (1,5).

Worked solution

a) Substitute x = 1, y = 5 into y = -x + k:

b) For y = 2x + 3 at x = 1: y = 2(1) + 3 = 5.

For y = -x + 6 at x = 1: y = -(1) + 6 = 5.

So (1, 5) lies on both lines.

c) The line through (0,0) and (1,5) has gradient m = (5-0)/(1-0) = 5. Equation: y = 5x.