Find b given that A (-6,2) B (b,0) and C (3,-4) are collinear
Answers (2)
in order for them to collinear, the slope from A to B must be equal to the slope from B to C, or the slope from C to B must be equal to the slope from B to A
let m1 = slope from A to B ; m2 = slope from B to C
m1 = (0-2)/[b-(-6)] = -2/(b+6)
m2 = (-4-0)/(3-b) = -4(3-b)
m1 = m2
-2/(b+6) = -4(3-b)
cross multiply
-2(3-b)=-4(b+6)
-6+2b = -4b - 24
2b + 4b = -24 + 6
6b = -18
divide both sides by 6
b = -3
A (-6,2) B (b,0) and C (3,-4) "collinear" means they are on the same line.
Slope is rise over run. Run is horizontal distance, left to right. Run is always positive because we always go left to right. Rise is the vertical change in that same distance. A negative rise means it drops. Line AC runs 9 units from -6 to 3 and rises -6 units from 2 to -4 so the slope is -6/9 = -2/3.
The equation of a line is y = mx where m is the slope. That line passes through the origin. If you want it to pass through some point (a, b) you subtract the coordinates like this: y - b = m(x - a). That is the point-slope form, and you can rewrite it in other forms if it is convenient. You may do this with any point on the line. They all reduce to the same equation. We will use (3, -4) this time.
y + 4 = (-2/3)(x - 3) Notice that we do not write -2/3(x - 3) because that would be ambiguous.
y + 4 = -2x/3 + 2
y = -2x/3 - 2 This is the slope-intercept form. Intercept is where the line crosses the y axis when x=0. Now you can plug in the given value of y and solve for x.
0 = -2x/3 - 2 Add 2.
2x/3 = -2 Multiply by 3.
2x = -6 Divide by 2.
x = -3 <-- ANSWER
Check: www.wolframalpha.com/input/?i=plot+y+%3D+-2x%2F3+-+2