Answer:
- (a) y = 5/2x +3
- (b) y -4 = -1/2(x -1) ⇒ y = -1/2x +9/2
- (c) Consistent; the lines have different slope
- (d) (x, y) = (1/2, 17/4)
Explanation:
(a) To make use of the forms of the equation for a line this problem requires, we must first calculate the slope (m) of the line between the two given points. That is computed by ...
... m = (change in y)/(change in x)
... m = (8 -3)/(2 -0) = 5/2
The second given point tells us the y-intercept (b) is 3, so the equation for the line is ...
... y = mx + b
... y = 5/2x + 3
(b) As in part (a), we must calculate the slope of the line. That is ...
... m = (change in y)/(change in x)
... m = (4 -3)/(1 -3) = 1/(-2) = -1/2
Using the first point (h, k) in the point-slope form of the equation of a line, we have ...
... y -k = m(x -h)
... y -4 = -1/2(x -1) . . . . . point-slope equation
Simplifying this, we get
... y = -1/2x +1/2 +4
... y = -1/2x +9/2 . . . . . slope-intercept equation
(c) The slopes of the two equations are different, so they must intersect at exactly one point. The equations are consistent.
(d) We can substitute the "y" from one equation into the other to give ...
... 5/2x +3 = -1/2x +9/2
... 3x = 3/2 . . . . . . . add 1/2x -6/2
... x = 1/2 . . . . . . . . divide by 3
Using the first equation to find y:
... y = (5/2)·(1/2) +3 = 5/4 +12/4
... y = 17/4
The point of intersection is (x, y) = (1/2, 17/4).