Given the plane representing (-4,8) and (6, -12):
Answers
Answer:
Explanation:
An important topic of high school algebra is "the equation of a line." This means an equation in x and y whose solution set is a line in the (x,y) plane.
The most popular form in algebra is the "slope-intercept" form
y = mx + b.
This in effect uses x as a parameter and writes y as a function of x: y = f(x) = mx+b. When x = 0, y = b and the point (0,b) is the intersection of the line with the y-axis.
Thinking of a line as a geometrical object and not the graph of a function, it makes sense to treat x and y more evenhandedly. The general equation for a line (normal form) is
ax + by = c,
with the stipulation that at least one of a or b is nonzero. This can easily be converted to slope-intercept form by solving for y:
y = (-a/b) + c/b,
except for the special case b = 0, when the line is parallel to the y-axis.
If the coefficients on the normal form are multiplied by a nonzero constant, the set of solutions is exactly the same, so, for example, all these equations have the same line as solution.
2x + 3 y = 4
4x + 6y = 8
-x - (3/2) y = -2
(1/2)x + (3/4)y = 1
In general, if k is a nonzero constant, then these are equations for the same line, since they have the same solutions.
ax + by = c
(ka)x + (kb)y = kc.
A popular choice for k, in the case when c is not zero, is k = (1/c). Then the equation becomes
(a/c)x + (b/c)y = 1.
Another useful form of the equation is to divide by |(a,b)|, the square root of a2 + b2. This choice will be explained in the Normal Vector section.
Exercise: If O is on the line, show that the equation becomes ax + by = 0, or y = mx.
Exercise: Find the intersections of this line with the coordinate axes.
Exercise: What is the equation of a line through (0,0) and a point (h,k)?