The key underlying idea to this is that we are able to plot data values on a graph (since we have the values of y and y^2 - x ready for plotting).

We mainly wish to see how the linear form of the curve appears to be like. Y and X can contain a mix of terms x and/or y as long as they are not raised to an unknown power.

Noted. Thank You

For linear law questions, you want to express the dependent variable with a linear relationship to that of the independent variable.


Hence, Y = mX + C

m is your gradient, C is your Y-intercept

So, for y² - ay = x - b

We realise that a and b are constants. They don't change.

What you want to do is to rearrange the equation in such a form where you can get it in the same format at Y = mX + C

Since b is a constant on its own, this is easy. We will use b as our 'C' and leave it as it is.


For ay, a is paired with y. This is already analogous to 'mX'

So just rearrange ay and x.


y² - x = ay - b


Comparing this to Y = mX + C,

y² - x = Y

y = X

a = m

-b = C


The Y does not strictly have to be in terms of the original y only.
Likewise, X does not strictly have to be in terms of the original x only.

Here's another way you can rearrange the equation :

y² - ay = x - b

Divide both sides by y,


y - a = x/y - b/y

y - x/y = -b/y + a

y - x/y = -b(1/y) + a

(Which we realise was just dividing both sides of 'y² - x = ay - b' by y)


Here,

X = 1/y
Y = y - x/y
m = -b
C = a

This is a different version but you still get a linear relationship.

The only caveat is, the graph is now undefined for y = 0. So if you have a data point where y = 0, then it won't work. It would cause a break in the line.