Equation of the line : y = mx + c
m is the gradient and c is the y-intercept

Since the intercepts at the axes are equal in magnitude but opposite in sign,

And since y-intercept is denoted as c, then the x-intercept = -c

Next, recall that at the y-intercept, x = 0
At the x-intercept, y = 0

So now we know the coordinates of three points that lie on the line are :

(0,c) , (-c,0) and (4,5)


Sub (-c,0) into the equation,

0 = m(-c) + c
0 = -cm + c
cm = c
m = 1

(We can divide both sides by c since c is clearly non-zero. If c were 0 then there would be no opposite sign for the intercepts since the line would pass through the origin. i.e both the y-intercept and x-intercept = 0)

Sub (4,5) and m = 1,

5 = 1(4) + c
5 = 4 + c
c = 1

Therefore, equation is:

y = x + 1

Alternatively, once we know (-c,0) and (0,c) are coordinates of two points on the line,

Gradient of line, m = (y2-y1)/(x2-x1)

= (c - 0) / (0 - (-c) )
= c / c
= 1

Then sub m = 1 and (4,5) accordingly.

We also can visualise it as such:

The line makes a right angled-isosceles triangle with the axes, since from the origin, the distance to the intercepts are the same.

The intercepts are equal in magnitude but opposite in sign so one must be positive and one must be negative. This tells us that the line has a upward slope.

It's either a positive y-intercept, negative x-intercept or the other way around.

Eg. (0,5) and (-5,0)
Eg. (0,-8) and (8,0)


This also means that between the two intercepts, the rise and run of the line are equal in magnitude and have same sign.


For example, when moving from (-c,0) to (0,c),

rise = c - 0 = c
run = 0 - (-c) = 0 + c = c

rise = run

rise/run = 1

We know the gradient must be positive.

If we move the other way, (0,c) to (-c,0)

rise = 0 - c = -c
run = -c - 0 = -c

Rise = run

c could be either positive or negative. In this question, it was found to be the former.

Thank you very much