Menu
The concepts of direct, inverse and joint variation are commonly discussed but linear relation is not discussed often. Today, we will discuss what we mean by “linearly related”. A linear relation is one which, when plotted on a graph, is a straight line. In linear relationships, any given change in an independent variable will produce a corresponding change in the dependent variable, just like a change in the 
-coordinate produces a corresponding change in the 
-coordinate on a line.
We know the equation of a line: it is 
, where 
is the slope and 
is a constant.
Let’s illustrate this concept with a GMAT question. This question may not seem like a geometry question, but using the concept of linear relations can make it easy to find the answer:
Question: A certain quantity is measured on two different scales, the 
-scale and the 
-scale, that are related linearly. Measurements on the -scale of 6 and 24 correspond to measurements on the -scale of 30 and 60, respectively. What measurement on the -scale corresponds to a measurement of 100 on the -scale?
(A) 20
(B) 36
(C) 48
(D) 60
(E) 84
Let’s think of the two scales 
and 
as – and -coordinates. We can get two equations for the line that depicts their relationship:
……. (I)
……(II)
(II) – (I)
Plugging 
in (I), we get:
Therefore, the equation is 
. Let’s plug in 
to get the value of :
Answer (C).
Alternatively, we have discussed the concept of slope and how to deal with it without any equations in our Co-ordinate Geometry Module. Think of each corresponding pair of and as points lying on a line – (6, 30) and (24, 60) are points on a line, so what will (
, 100) be on the same line?
We see that an increase of 18 in the -coordinate (from 6 to 24) causes an increase of 30 in the -coordinate (from 30 to 60).
So, the -coordinate increases by 
for every 1 point increase in the -coordinate (this is the concept of slope).
From 60 to 100, the increase in the -coordinate is 40, so the -coordinate will also increase from 24 to 
.
Again, (C) is our answer.