What must be true about the solution to this system of equations?

x+4y+9z = -96
-2y+9z = -47
z = -7

a) The solution will describe the intersection point(s) of the 3 graphs.

b) The value of z must be -7.

c) The solutions will have values for x, y, and z.

d) All of the above statements are true.

I initially chose D.

I agree with D.

To determine the correct answer, let's analyze the system of equations:

x + 4y + 9z = -96 ---(1)
-2y + 9z = -47 ---(2)
z = -7 ---(3)

To find the solution to this system, we can use the method of substitution. Since the value of z is given in equation (3), we can substitute it into equations (1) and (2).

Substituting z = -7 into equation (1), we get:
x + 4y + 9(-7) = -96
x + 4y - 63 = -96
x + 4y = -96 + 63
x + 4y = -33 ---(4)

Substituting z = -7 into equation (2), we get:
-2y + 9(-7) = -47
-2y - 63 = -47
-2y = -47 + 63
-2y = 16
y = -8 ---(5)

Now that we have found the value for y, we can substitute it back into equation (4) to find the value of x:
x + 4(-8) = -33
x - 32 = -33
x = -33 + 32
x = -1 ---(6)

So, the solution to the system of equations is x = -1, y = -8, and z = -7.

Now let's evaluate the given statements one by one:

a) The solution will describe the intersection point(s) of the 3 graphs.
By solving the system of equations, we found a unique solution for x, y, and z. Since the solution is unique, it represents a single point of intersection for the three graphs. So, statement a) is true.

b) The value of z must be -7.
By equation (3), z is given as -7. This is the only value that satisfies equation (3). So, statement b) is true.

c) The solutions will have values for x, y, and z.
By solving the system of equations, we found values for x, y, and z. So, statement c) is true.

d) All of the above statements are true.
After evaluating each statement, we find that all of them are true. Therefore, the correct answer is d) All of the above statements are true.

So, your initial choice of D is the correct answer.