The equation of a straight line - Ty Harness 24th April 2005

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The equation of a straight is the first lesson ever taught in geometry.

Using a conventional set of Cartesian XY axes, a straight line can be described with the formula:

y = mx + c

where

y is any ordinate parallel to the X axis. m is the slope,rate or gradient of the line. Basically the ratio of how y changes with x. The above graph shows the change in y = y1 - y0 = 14 which is also known as delta y. Again the change in x = x1 - x0 = 24 and known as delta x

m = delta y / delta x = 14/24 = 0.58333333

The very nature of a straight line is that it's linear and constant so the gradient is same any where measured. The choice of x0,y0 and x1,y1 is purely arbitrary.

x is any ordinate parallel to the Y axis.

c is y value where x=0. c= +6 for the above example.

y is referred to as a function of x, y=f(x)

f(x) = mx + c

If you have not seen or discussed the concept of a function then we can exchange x for any numeric value in order to calculate the corresponding y value.

P has the co-ordinate 66,y

y = f(66) = 0.58333*66 + 6 = 44.5

The equation of a straight line is used for everything. Any geometry text book will give you half a dozen examples and a 100 test questions to practice on.

My first example was a spring. You load a spring and measure the extension of the spring. Plot load verses extension on a graph and repeat the procedure with a few different loads and the points all seem to pass through a straight line. I could conclude that the spring was highly predictable.

Another example is the adjustment of flying time in pigeon racing where a member's clock may run fast or slow against true time. A slow clock would make the pigeon appear faster. i.e. covering a distance in less time and the converse where a fast clock would be a disadvantage. We make the assumption that a clock running either fast or slow does so at a constant rate. Using the equation of a straight we can predict the true time at which the bird got home. That's the only fair way, hence every bird must be racing against the true time to be comparable.

You can see some worked examples at Pigeon Calculator software

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