Meaning of the constant m in y = mx.

Inclination of a straight line

Setting in the equation y=mx the dependent variable x=1, x=2, x=3, . . . , yields 

As you travel along the x-axis by equal distances, the value of y changes by the same amount. The uniform growth of the variable y is obvious from the figures in the last section.

By the rise of a railway track or road, shown by the triangle below, one understands, in general, the ratio PQ/OQ (cf. also Proportions). If you transfer this definition to the present case, then

P₁Q₁/OQ₁ = m/1 = m; P₂Q₂/OQ₂ = m/1 = m , etc. or also
P₂Q₂/OQ₂ = m/2 = m; P₃Q₃/OQ₃ = 3m/3 = m , etc.

In fact, the equation y = mx tells that we must have y/x = m for every corresponding pair of values. The rise is everywhere the same.

m represents the gradient of the straight line.

The larger m, the steeper is the line; moreover, the angle , which the line forms with the abscissa, must be sharp for positive m and obtuse for negative m (y = -x is shown by the figure below). In the negative case, the line does not pass any more through the I. and III. quadrants, but through the II. and IV. quadrants. If m is negative, then 90º <  < 180º.

The graph below shows straight lines with different inclines. The smallest value, which m can assume, is m = 0. If y = 0·x, then y is identically zero: y  0, that is, you obtain the abscissa for which we can now write the equation y = 0.

The graph of the equation y=0 is the abscissa.  

As you let the positive m increase, the lines y = mx approach more and more the ordinate axis without ever reaching it. It does not make sense to speak here of m becoming infinite! You cannot obtain the ordinate axis in this way. However, for every point of it, you must have x = 0. This condition is fulfilled by the equation x = 0·y = 0 - The image of the equation x=0

m as factor of proportionality

Since y = mx can be rewritten y/x = m, all values, which belong together, are in the same ratio:

y₁/x₁= y₂/x₂ = y₃/x₃ = . . . = m.

If the ratio y:x is equal for all pairs of values, you say that y is proportional to x and call m the factor of proportionality.

In many cases, the value of m is unimportant. If you want to obtain a second pair of values which belong together, you start out from the proportion y₁/x₁= y₂/x₂; with its help, you can find readily from three given values the fourth value. It is here unimportant, whether the proportion is as shown or the inner and outer terms have been interchanged: y₁/y₂ = x₁/x₂.

Example 1: Your capital yields at 4 % interest 17 Baht. How much will you receive from the same capital in the same time at 5%?

As the interest rate becomes larger, the same capital yields more interest, that is, the interest rate and interest paid are proportional to each other. 

Example 2: Convert 12' (x) into degrees (y)!

You know that 1° has 60', whence 

Exercise:

Five litres of Diesel cost 60 Baht.What is the cost of 13.5 litres?

Answer

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