Parameterization! Math Master's Please Help Me?

Question

Consider the lines parametrized by

x(t) = 3t-7 and x(t) = 5t + 6
y(t) = 4-9t y(t) = ct+ 8

b) For what value of c if any, do these lines intersect at (5, -32)?
c) For what value of c if any, are these lines the same?

Answer 1

Which x(t) goes with which y(t) ?

Once you have P1(x(t),y(t)) and P2(x(t),y(t))

b) set P1 = P2 = (5,-32) and see if there's a solution.

Answer 2

It’s not quite clear. IF
Line#1 being x=3t-7, y=4-9t;
Line#2 being x=5t+6, y=ct+8; THEN
For line#1. Does [5,-32] belong ? 5=3t-7, -32=4-9t; t=4; Yes! For both x and y.
For line#2. Does [5,-32] belong ? 5=5t+6, t=-1/5; -32=ct+8, hence c=(-32-8)/(-1/5) = 200; OK!

Now if they are the same then their derivatives must be the same (1).
For line#1 dx = 3dt, dy = -9dt, dy/dx=-3;
For line#2 dx = 5dt, dy = cdt, dy/dx=c/5=-3; so c=-15;
And they must have all points common for both (2).
Suppose t=0 for line#1, then x=-7, y=4; plug x and y into line#2;
-7=5t+6, 4=-15t+8; 5t=-13, 15t=4; t=-13/5, t=4/15; No! for c=-15 they are only parallel!