Find the point at which the line <1,-1,4>+t<3,4,-1> intersects the plane 4x-y-z=73.

What process do i take to solve this problem?

To find the point at which the line intersects the plane, you can follow these steps:

1. Set up the equations for the line and the plane.
Line equation: x = 1 + 3t, y = -1 + 4t, z = 4 - t
Plane equation: 4x - y - z = 73

2. Substitute the line equations into the plane equation.
4(1 + 3t) - (-1 + 4t) - (4 - t) = 73

3. Simplify the equation.
4 + 12t + 1 - 4t - 4 + t = 73

4. Combine like terms.
9t - 3 = 73

5. Solve for t.
9t = 76
t = 76/9

6. Substitute the value of t back into the line equation to find x, y, and z.
x = 1 + 3(76/9)
y = -1 + 4(76/9)
z = 4 - (76/9)

7. Simplify the values of x, y, and z.
x ≈ 9.22
y ≈ 34.67
z ≈ -1.78

Therefore, the point of intersection between the line and the plane is approximately (9.22, 34.67, -1.78).

To find the point at which the line intersects the plane, we need to solve the system of equations formed by the line and the plane. Here are the steps to solve this problem:

Step 1: Write the equation for the line in parametric form.
Given the line <1, -1, 4> + t<3, 4, -1>, we can write the parameterized equation for the line as:
x = 1 + 3t
y = -1 + 4t
z = 4 - t

Step 2: Substitute the equations for x, y, and z from the line into the equation of the plane.
The equation of the plane is given as 4x - y - z = 73. Replace x with (1 + 3t), y with (-1 + 4t), and z with (4 - t):
4(1 + 3t) - (-1 + 4t) - (4 - t) = 73

Step 3: Simplify the equation and solve for the parameter 't'.
Solving the equation will give you the value of 't', which represents the position of the point of intersection on the line.

4 + 12t + 1 - 4t + 4 - t = 73
9t + 9 = 73
9t = 64
t = 64/9

Step 4: Substitute the value of 't' back into the parametric equations of the line to find the coordinates of the point of intersection.
Substituting the value of 't' back into the parametric equations will give us the coordinates of the point.

x = 1 + 3(64/9) = 217/9
y = -1 + 4(64/9) = 255/9
z = 4 - 64/9 = 176/9

Therefore, the point of intersection of the line and the plane is (217/9, 255/9, 176/9).