We received correct solutions from Ash and Lucy, two students at Tiffin Girls' School. Well done to you both.
Ash sent us this correct solution:

The gradient of the first line must be the negative reciprocal of the gradient of the other line:
if you multiply the two gradients you always get -1.

For example:
Take a random gradient, say
4
7


The negative reciprocal gradient will be
-7
4

Make up two equations with these gradients, say

y = 4x
7

and
y = -7x
4

Draw them on a grid

perpendicular lines
You get perpendicular lines.


Lucy recorded how she worked through this problem:

y = x is perpendicular to y = -x

y = x is perpendicular to y = -x + 2
(the y-intercept doesn't affect the gradient of the line)

y = 2x is perpendicular to y = -x/2

y = -3x is perpendicular to y = x/3
(to make a line perpendicular you need to invert the gradient, or take the reciprocal, and change the sign)

y = -2x is perpendicular to y = x/2

I can see a pattern here: when the two gradients of perpendicular lines are multiplied together they give -1, and the y-intercept does not affect if the line is perpendicular or not.

I will now try to work out what the perpendicular line of some other lines will be using this formula:


y = 7x - 3
Using my formula I predict that a line which is perpendicular to this line will be y = -x/7 - 3

perpendicular lines
When I tested the lines out, I found that the formula had worked.


y = x/3 + 4
Using my formula I predict that a line which is perpendicular to this line will be y = -3x + 4

perpendicular lines
Having drawn out the lines I found that the formula worked and the lines were perpendicular.


y = -7x/3 + 2
I predict that a line which is perpendicular to this line will be y = 3x/7

perpendicular lines
From drawing out these lines I can see that they are perpendicular.