Question

A system of equations and a matrix.Short description, A system of equations and a matrix.,Long description,
A system of equations with 3 rows: 1st row 2 X minus 4 Y minus 2 Z equals 0. 2nd row 0 X plus 1 Y plus 4 Z equals 0. 3rd row 3 X minus 6 Y minus 3 Z equals 0.
A 3 by 4 matrix: First row first column 1, second column 0, third column 7, fourth column 0. Second row first column 0, second column 1, third column 4, fourth column 0. Third row first column 0, second column 0, third column 0, fourth column 0.

Question
What is the solution?

Answer options with 5 options
A.
the point negative 1 z comma 7 z comma 4 z

B.
the point negative 7 z comma negative 4 z comma 1 z

C.
the point 4 z comma 7 z comma 1 z

D.
the point negative 1 z comma negative 7 z comma negative 4 z

E.
the point negative 4 z comma 1 z comma negative 7 z

Answer 1

To solve the given system of equations, let's first rewrite the equations clearly. The three equations represented by the rows of the system are:

1. \( 2X - 4Y - 2Z = 0 \)
2. \( Y + 4Z = 0 \)
3. \( 3X - 6Y - 3Z = 0 \)

We can simplify the third equation since it is a scalar multiple of the first equation, indicating that it does not provide new information. Therefore, we consider the first two equations only:

1. \( 2X - 4Y - 2Z = 0 \) (Equation 1)
2. \( Y + 4Z = 0 \) (Equation 2)

From Equation 2, we can express \( Y \) in terms of \( Z \):
\[ Y = -4Z \]

Now, substituting \( Y \) into Equation 1:
\[
2X - 4(-4Z) - 2Z = 0
\]
\[
2X + 16Z - 2Z = 0
\]
\[
2X + 14Z = 0
\]
Thus,
\[
2X = -14Z
\]
\[
X = -7Z
\]

Now we can summarize our variables in terms of \( Z \):
- \( X = -7Z \)
- \( Y = -4Z \)
- \( Z = Z \) (parameter)

Thus, the solution set can be expressed in vector form:
\[
(X, Y, Z) = (-7Z, -4Z, Z)
\]

This corresponds to option B:

**B. the point negative 7 z, negative 4 z, 1 z**

So, the answer is **B**.