Linear Relations

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: 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? (A) 20 (B) 36 (C) 48 (D) 60 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. 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.