First suppose the line parallel to the axis of x. Let BC be the line meeting the axis of y in B; suppose OB=b.

Since the line is parallel to the axis of x, the ordinate PM of any point of it is equal to OB. Hence calling y the ordinate of any point P, we have for the equation to the line y=b.

Next suppose the line parallel to the axis of y. Let AD be the line meeting the axis of x in A; suppose OA=a. Since the line is parallel to the axis of y, the abscissa of any point of it is OA. Hence calling x the abscissa of any point, we have for the equation to the line

16. We have thus proved that any straight line whatsoever is represented by an equation of the first degree; we shall now shew that any equation of the first degree with two variables represents a straight line.

The general equation of the first degree with two variables is of the form

Ax+By+C= 0........

A, B, C being finite or zero.

.(1),

First suppose B not zero; divide by B, then from (1)

Now we have seen in Art. 14, that if a line be drawn

C

meeting the axis of y at a distance from the origin

B

and making with the axis of x an angle of which the tangent A

is, then (2) will be the equation to this line. Hence (2),

B'

and therefore also (1), represents a straight line.

If A=0, then by Art. 15 the line represented by (1) is parallel to the axis of x.

If B=0, then (1) becomes

Ax+ C = 0,

and from Art. 15 we know that this equation represents a line parallel to the axis of y.

Hence the equation Ax + By + C = 0 always represents a straight line.

17. Equation in terms of the intercepts. The equation to a line may also be expressed in terms of its intercepts on the

Let A and B be the points where the straight line meets the axes of x and y respectively. Suppose OA=a, OB=b. Let P be any point in the line; x, y its co-ordinates; draw PM parallel to OY. Then by similar triangles,

18. It will be a useful exercise for the student to draw the straight lines corresponding to some given equations. Thus suppose the equation 2y+3x=7 proposed; since a straight line is determined when two of its points are known, we may

find in any manner we please two points that lie on the line, and by joining them obtain the line. Suppose then x=1, it follows from the equation that y=2; hence the point which has its abscissa = 1 and its ordinate = 2 is on the line. Again, suppose x=2, then y; the point which has its abscissa = 2 and its ordinate is therefore on the line. Join the two points thus determined and the line so formed, produced indefinitely both ways, is the locus of the given equation. The two points that will be most easily determined are generally those in which the required line cuts the axes. Suppose x=0 in the given equation, then y=1, that is, the line passes through a point on the axis of y at a distance from the origin. Again, suppose y = 0, then x=}, that is, the line passes through a point on the axis of x at a distance from the origin. Join the two points thus determined, and the line so formed, produced indefinitely both ways, is the locus of the given equation. What we have here ascertained as to the points where the line cuts the axis, may be obtained immediately from the equation; for if we write it in the form

and compare it with the equation in Art. 17,

Again, suppose the equation y=x proposed. Since this equation can be satisfied by supposing x=0 and y = 0, the origin is a point of the line which the equation represents; therefore we need only determine one other point in it. Suppose x=1, then y = 1; here another point is determined and the line can be drawn. The line may also be constructed by comparing the given equation with the form in Art. 14, y = mx.

This we know represents a line passing through the origin and making with the axis of x an angle of which the tangent Hence y=x represents a line passing through the origin and inclined at an angle of 45° to the axis of x.

is m.

Similarly the equation y=-x represents a line inclined to the axis of x at an angle of which the tangent is -1; that is, at an angle of 135°. Hence this equation represents a line through O bisecting the angle between OY and OX produced to the left in the figure to Art. 14.

19. The student is recommended to make himself thoroughly acquainted with the previous Articles before proceeding with the subject. In Algebra the theory of indeterminate equations does not usually attract much attention, and the student is sometimes perplexed on commencing a subject in which he has to consider one equation between two unknown quantities, which generally has an infinite number of solutions.

Our principal result up to the present point is, that a straight line corresponds to an equation of the first degree, and the student must accustom himself to perceive the appropriate line as soon as any equation is presented to him. The line can be determined by ascertaining two points through which it passes, that is, by finding two points such that the co-ordinates of each satisfy the given equation, and the line being thus determined, the co-ordinates of any point of it will satisfy the given equation.

20. Equation to a straight line in terms of the perpendicular from the origin, and the inclination of this perpendicular to the

axis.

B

R

M

Let OQ be the perpendicular from the origin 0 on a line AB. Take any point P in the line; draw PM perpendicular to OA, MN perpendicular to OQ, and PR perpendicular to MN. Suppose OQ=p, and the angle QÕA=a. Let x, y be the co-ordinates of P; then

21. We have given separate investigations of the different forms of the equation to a straight line in Articles 14, 17, 20; any one of these forms may however be readily deduced from either of the others by making use of the relations which exist between the constant quantities. The quantity which we have denoted by b in Art. 17, that is OB, is denoted by c in Art. 14;

in Art. 14 we have denoted the tangent of BAX by m,

In Art. 20, OA cos a = OQ, and OB sin a = OQ; that is,

p=a cos a = b sin a.......................

therefore from (2) and (3), m = - cot a........

Also if the equation

T. C. S.

Ax+ By + C = 0,

2