Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

166

ANALYTIC GEOMETRY

Since the lines (1) and (2) intersect in the point (1, 1), the
coordinates of this point make the left-hand sides of equations
(1) and (2), namely, the expressions,
x + y -2 and x - y,
vanish. Consequently, they always make the left-hand side
of equation (3) vanish. In other words, equation (3) is satisfied
by the coordinates of the point of intersection of the lines (1) and
(2), NO MATTER WHAT VALUE k HAS. This means that all the
straight lines represented by (3) go through the point of intersection of the lines (1) and (2).
The result can be restated in the following form. Let the
single letter u stand for the whole expression x + y - 2:
u= x + y -2,
the sign meaning identically equal, i.e. equal, no matter what
values x and y have. Similarly, let v stand for x- y:
v x -y.
Then (3) takes on the form:
(4) u + cv= 0.
We now restate our result.
If u = 0 and v = 0 are the equations of two intersecting straight
lines, then the equation
u + kv = O
represents a straight line which goes through the point of intersection of the two given lines.
By giving to k a suitable value, u + kv = 0 can be made to
represent any desired line through the point of intersection
(x1, y,) of the given lines, with the sole exception of the line
v = 0. For, let L be the desired line, and let (x2, Y2) be a
point of L distinct from (x1, yO). Then, on substituting for x
and y the values x2 and y2 in the equation u + kv = 0, we obtain an equation, in which k is the unknown. This equation
can be solved for k, since v does not vanish for the point
(x2, Y2).