(8) Find the coördinates of the four corners of the figure formed by the lines (1), (2), (6), and (7).

(9) Find the coördinates of the four corners of the figure formed by the lines (2), (4), (6), and (7).

THE GRAPH OF THE LINE y = mx + b

245. In answering the question, what is the geometrical figure represented by y = mx+b, begin with the simplest case first.

Here, on giving different values to a, one gets for

1. y = x.

A little observation and reflection will convince one that the

points O, P, P2, P3,

tion given in Fig. 4.

etc., and P', P", P'"', etc., will have the loca

All other sets of values x and y which satisfy

the equation y=x will represent points situated on the straight line P' OP1

For example, x = 18, y=1 is the point Q.

2. y = mx, where m is any number.

The points corresponding to the values x = 0, 1, 2, etc, .. and -1, -2, -3, etc., substituted in y = mx are:

On comparing the corresponding values of x and y in 1 with

But, according to a geometric principle, if 0, P1, P., etc., lie on a straight line, then O, L, L, etc., lie on a straight line.

NOTE-In case a pupil has not had this geometric principle, it will do no harm to assume the fact for the present, as this is the only assumption that will have to be made in connection with this subject, and he will soon have enough geometry to understand this proof

Therefore, y = mx also represents a straight line through the origin 0, the line L'OL1, Fig. 4.

3. y = mx + b. For the same values of x used in 2, the following points are determined:

R

R"

x= =0

Sx=1 1

y=b=OR; R1 { y=m+b=M,L,+OR=M,L,+L,R,=M ̧R ̧;

( x = 2

| y= −2m+b=—N"L"+OR=—N"L" +R"L" — — N"R".

L1R1

....

All the points R, R, R", etc., are located at the same distance, b = OR = = L'R". measured along the vertical lines through 0, L1, N", etc., and lie in the line R"RR1, parallel to the line L'OL,.

An important property of the line y =mx+b is to be noted, namely, that it cuts off the intercept OR b on the positive portion of the y-axis. If b were negative, the line would have the position A, B, C, cutting off the negative intercept b=0B on OY'.

246. The Solutions of y=mx+b.-It follows from the previous section that the infinite sets of solutions which x and y can have in the equation y = mx + b represent an infinite number of points distributed along the straight line R"RR,; for example,

247. The Intercepts. It has already been noted that the intercept of the line y=mx+b on the y-axis is + b = OR. It is found by making the variable point on the line y = mx + b move along the line till it falls on OY, and this will happen when x = 0, and the corresponding value of y, namely, y = + b, is the intercept, OR, of the line on OY.

Now make this variable point move along the line y = mx + b, i. e., along R"RR,, until it falls on O. at N, where y = 0; then the corresponding value of x will be found from the equation 0 = mx + b, i. e., x = which is the intercept of the line RRR, on the axis XOX. If is calleda, the equation y = mx + b may be written

(1)

m

b

m

b

248. The rule for the signs of the intercepts on the axes will be: Intercepts on OX' and OY' are negative and those on OX and OY are positive.

represent lines crossing the second, first, third, and fourth angles respectively, as shown in Fig. 5.

INTERSECTIONS OF PAIRS OF LINES

249. 1. The solution of equations (1) and (2), 248, is x =

=

51

and y 2. Since these values of x and y satisfy equations (1) and (2), they are the coördinates of the point of intersection of (1) and (2), namely, P1, which is shown in Fig. 6.

2. The solution of (2) and (3) is x =

3

=

This

point P1, which is the intersection of lines (2) and (3), Fig. 6. 3. The solution of (3) and (4) is x, y = is point P, the intersection of (3) and (4), shown in Fig. 6. 4. The solution of (4) and (1) is x = 62, y = the coordinates of the point P, the point common to lines (1) and (4), Fig. 6.

, which are

250. RESUME Thus is reached the beautiful geometrical representation of an equation in one and two unknown quantities; a conditional equation in one unknown quantity is in every case represented by a line parallel to the x- or y-axis, either on the positive or the negative side of the axis, 244, II; an equation in two unknown. quantities (y = mx + b) is represented by a straight line, 245, 3; and, when it is written in the form of + Y = 1, a and b are respectively the intercepts of the line on the x- and y-axes, 247. If both intercepts are positive the line crosses the first angle XOY; if a is and b is, the line crosses the second angle; if both a and

а b