Find the general form of the equation of the plane passing through the three points.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (-4,4,-1), (1,6,2), (-5,-3,6)
A)35x -38y -33z = 0
B)4x -4y 1z = 0
C)4x -4y 1z + 259 = 0
D)35x -38y -33z + 259 = 0
E)35x -38y -33z - 259 = 0

Question

Quizplus · Accepted Answer

To find the equation of the plane passing through three points, we can use the formula:
$$
Ax + By + Cz = D
$$
where A, B, C, and D are constants that we need to determine. 
To find these constants, we can use the three points given to create three equations. For example, using point (-4, 4, -1), we get:
$$
A(-4) + B(4) + C(-1) = D
$$
which simplifies to:
$$
-4A + 4B - C = D
$$
Doing this for the other two points gives us a system of equations:
$$
\begin{cases}
-4A + 4B - C = D \
A + 6B + 2C = D \
-5A - 3B + 6C = D 
\end{cases}
$$
We can solve this system using Gaussian elimination or other methods. The resulting constants are:
$$
A = 35, B = -38, C = -33, D = -259
$$
By dividing all the coefficients by -259, we can reduce the equation to:
$$
-\frac{35}{259}x + \frac{38}{259}y + \frac{33}{259}z = 1
$$
Multiplying both sides by -1 gives us the equation in the form given in the answer, choice D.

Find the General Form of the Equation of the Plane