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 x-coordinate produces a corresponding change in the y-coordinate on a line.
We know the equation of a line: it is y = mx + c, where m is the slope and c 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 R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale?
Let’s think of the two scales R and S as x- and y-coordinates. We can get two equations for the line that depicts their relationship:
30 = 6m + c ……. (I)
60 = 24m + c ……(II)
(II) – (I)
30 = 18m
m = 30/18 = 5/3
Plugging m = 5/3 in (I), we get:
30 = 6*(5/3) + c
c = 20
Therefore, the equation is S = (5/3)R + 20. Let’s plug in S = 100 to get the value of R:
100 = (5/3)R + 20
R = 48
48 (answer choice C) is our answer.
Alternatively, we have discussed the concept of slope and how to deal with it without any equations in our Co-ordinate Geometry module here. Think of each corresponding pair of R and S as points lying on a line – (6, 30) and (24, 60) are points on a line, so what will (r, 100) be on the same line?
We see that an increase of 18 in the x-coordinate (from 6 to 24) causes an increase of 30 in the y-coordinate (from 30 to 60).
So, the y-coordinate increases by 30/18 = 5/3 for every 1 point increase in the x-coordinate (this is the concept of slope).
From 60 to 100, the increase in the y-coordinate is 40, so the x-coordinate will also increase from 24 to 24 + 40*(3/5) = 48.
Again, C is our answer.
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