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Given a straight line, we denote the angle between this line and the x–axis by j and the interval cut out by the line on the x-axis by b

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у М(х;у)

y-b

B j

b N

0 x

 

Definition 1. The slope tangent of the angle between a straight line and the x –axis is called the slope of the line and denoted by

k =tan j.

Suppose that k is the slope of a line and b is its y –intercept. Let us write an equation of this line.

Take a point M (x; y) and consider the triangle D BMN, where for any point М of the straight line under consideration. We obtain

, where and , whence .

Thus, the equation of a straight line with a slope has the form

.

The equation of a straight line with given slope passing through a given point. Suppose that a straight line passes through a point М0(x0,y0) and has slope k.

By analogy with the equation of a straight line with a slope consider the triangle М0MN; we have for any point М on the under

consideration or .

Thus, the required equation is

y – y 0 = k (x–x 0).

The equation of a straight line passing through two points. Suppose that a straight line passes through two points М 1(х 1; у 1) and М 2(х 2; у 2).

 

Take a point M(x,y) on the line and consider the vectors

and .

These two vectors и lie on the same straight line and are collinear.

The collinearity condition is the proportionality of the perspective coordinates, i.e.,

This is the equation of a straight line passing through the two given points.


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