Question

Find the point(s) of intersection (if any) of the plane and the line. Also determine whether the line lies in the plane. $$ 5 x+3 y=17, \quad \frac{x-4}{2}=\frac{y+1}{-3}=\frac{z+2}{5} $$

Short answer

The line intersects the plane at the point (4,-1,-2). The line does not lie within the plane.

Step-by-step solution

Rewrite the line equation

Rewrite line equation in terms of a parameter t: \(x = 2t+4\), \(y = -3t-1\), \(z=5t-2\).

Substitute x and y into the plane equation

Substituting x and y from the line equation into the plane equation gives: \(5(2t+4) +3(-3t-1)=17\).

Solve for t

Simplify and solve the equation for t: \(10t+20-9t-3=17\). This simplifies to \(t=0\).

Find the point of intersection

Substitute \(t=0\) into the line equation to get the point of intersection: \((x, y, z) = (2(0)+4, -3(0)-1, 5(0)-2) = (4,-1,-2)\).

Verify if the line lies in the plane

This process will turn into a trivial task as our line only intersects the plane at one point, thus the line does not lie within the plane.