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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.