Reduction to Linear Form

This section looks at reduction to linear form.

In order to show that data from a science experiment fits a rule, we have to be able to plot two variables on a graph so that a straight line relationship results (this will be the line of best fit if the data is experimental).

In questions, you may be given some data and you may be asked to plot y against 1/x . If the data lies in a straight line, you know, from the equation of a straight line (y = mx + c), that y = m/x + c, since 1/x is on the x-axis. You have therefore found the relationship between the data.

Example

V     5     10     15      20     25

R    149  175   219   280   359

By drawing a graph, show that these pairs of values may be regarded as approximations to values satisfying a relationship of the form R = a + bV² .

In this question, you would plot R against V² . If a straight line comes out, then you know that (from y = mx + c) R = mV² + c (which is just the same as R = a + bV², since m and c, a and b are just constants).

The second part of such a question may ask you to calculate the values of a and b. In this example, b is the gradient of the graph and a is the y-intercept (this can again be established by comparing the equation to y = mx + c), so you can read these values off from the graph.