It is known that the slope of the straight line L is k, and the straight line can be found after passing through the point P 1(x 1, y 1). How to find the equation of straight line L (Figure 1-24)?

The set point P(x, y) is any point on the straight line L different from P 1, which is obtained according to the slope formula of the two points.

Note the difference between equation (1) and equation (2): the coordinates of point P 1 do not satisfy equation (1), but satisfy equation (2). Therefore, the point P 1 is not on the graph represented by equation (1), but on the graph represented by equation (2).

By repeating the above process, it can be proved that the coordinates of each point on a straight line are the solutions of this equation. Through the inverse calculation of the above process, it can be proved that all the points whose coordinates are the solution of this equation are on the straight line L, so this equation is an equation with the slope of the straight line L being k and passing through the point P 1.

This equation is determined by a point on a straight line and the slope of the straight line, which is called the point inclination of the straight line equation.

When the slope of the straight line is 0 (figure 1-25), k=0, and the equation of the straight line is y = y 1.

When the slope of the straight line is 90 (Figure 1-26), the slope of the straight line does not exist, and its equation cannot be expressed by point inclination. However, because the abscissa of each point on L is equal to x 1, its equation is X = X 1.

(2) oblique cutting type

Given that the intercept of a straight line L on the Y axis is b and the slope is b, find the equation of the straight line.

This problem is equivalent to a point (0, b) on a given straight line and the slope k of the straight line. The equation for finding the straight line is a special case of the point oblique equation, and it can be obtained by substituting it into the point oblique equation:

y-b=k(x-0)

namely

The above equation is called linear oblique equation. Why is it called oblique equation? Because it is determined by the slope of the straight line and its intercept on the y axis.

When k≠0, the oblique equation is represented by a straight line, so the geometric meaning of k and b in the linear function is to represent the slope of the straight line and the intercept on the Y axis respectively.

(3) Two-point type

It is known that the positions of two points P 1(x 1, y 1), P2(x2, y2) and (x 1≠x2) on the straight line L are certain, that is, the equation of the straight line is solvable. Please find the equation of straight line L.

When y 1≠y2, in order to facilitate memory, we rewrite the equation as follows

Please name this equation: this equation is determined by two points on a straight line, which is called the two-point formula of a straight line.

The two-point equation should pay attention to the following two points: (1) equation is only applicable to straight lines that are not parallel to the coordinate axis, and when the straight lines are parallel to the coordinate axis (x 1=x2 or y 1=y2), the equation can be written directly; (2) To remember the two-point equation, you only need to remember the left side, and you can change Y to X when you see it on the right. The rules of foot code are exactly the same.