alanforde800Maximus
In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0). If (a,b) is the point of intersection of line l and line m, which of the following is true?
a) a > 0 and b > 0
b) a > 0 and b < 0
c) a < 0 and b > 0
d) a < 0 and b < 0
e) a = 0 or b = 0
We can begin by finding an equation for each line:
Line l:
slope = (0-2)/(1-0) = -2, y-intercept = (0, 2) = 2
Thus, an equation for line l is y = -2x + 2
Line m:
Slope = (0-(-4)/(4-0) = 1, y-intercept = (0, -4) = -4
Thus, an equation for line m is y = x - 4.
Now, let’s find their intersection by setting the right hand side of the two equations equal to each other:
-2x + 2 = x - 4
-3x = -6
x = 2
Substitute x = 2 back to either equation (let’s take the one for line m), we have:
y = 2 - 4 = -2
We see that the point of intersection is (2, -2), so a = 2 > 0 and b = -2 < 0.
Alternate Solution:
Using the given points, make a quick sketch of the two lines on the same set of coordinate axes. Note that line l has a negative slope and line m has a positive slope. Their point of intersection will be in Quadrant 4, where x is always positive and y is always negative.
Answer: B