Prove that the line {p:p = (1, 3, -1) + t(0, 3, 5)} lies entirely in the plane

Question

{(x, y, z): 2x – 5y + 3z = -16}

Answer 1

To prove that the line lies entirely in the plane, we need to show that every point on the line satisfies the equation of the plane.

Given the line {p: p = (1, 3, -1) + t(0, 3, 5)} and the plane {(x, y, z): 2x - 5y + 3z = -16}, we want to show that for any value of t, the point p = (1, 3, -1) + t(0, 3, 5) satisfies the equation 2x - 5y + 3z = -16.

Let's substitute the values of x, y, and z using the equation for the line:

x = 1 + 0t = 1
y = 3 + 3t
z = -1 + 5t

Now, let's substitute these values into the equation of the plane:

2(1) - 5(3 + 3t) + 3(-1 + 5t) = -16
2 - 15 - 15t - 3 + 15t = -16
-16 = -16

As we can see, the equation holds true for any value of t. Therefore, every point on the line lies in the plane. Hence, the line lies entirely in the plane {(x, y, z): 2x - 5y + 3z = -16}.