To determine which system of equations corresponds correctly to the given matrix equation, let's analyze the structure of the matrix equation given:
[tex]\[
\begin{pmatrix}
5 & 2 & 1 \\
7 & -5 & 2 \\
-5 & 3 & 1
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix} =
\begin{pmatrix}
16 \\
3 \\
12
\end{pmatrix}
\][/tex]
This matrix equation can be broken down into a system of linear equations, where each row of the matrix represents one equation. Specifically:
1. The first row [tex]\([5, 2, 1]\)[/tex] multiplied by [tex]\([x, y, z]^\top\)[/tex] gives the first equation:
[tex]\[
5x + 2y + z = 16
\][/tex]
2. The second row [tex]\([7, -5, 2]\)[/tex] multiplied by [tex]\([x, y, z]^\top\)[/tex] gives the second equation:
[tex]\[
7x - 5y + 2z = 3
\][/tex]
3. The third row [tex]\([-5, 3, 1]\)[/tex] multiplied by [tex]\([x, y, z]^\top\)[/tex] gives the third equation:
[tex]\[
-5x + 3y + z = 12
\][/tex]
So our system of linear equations is:
[tex]\[
\begin{aligned}
5x + 2y + z &= 16 \\
7x - 5y + 2z &= 3 \\
-5x + 3y + z &= 12
\end{aligned}
\][/tex]
We now compare this system with the given options:
- Option A:
[tex]\[
\begin{aligned}
5x + 7y - 5z &= 12 \\
2x - 5y + 3z &= 3 \\
x + 2y + z &= 16
\end{aligned}
\][/tex]
Clearly, the equations do not match.
- Option B:
[tex]\[
\begin{aligned}
5x + 7y - 5z &= 16 \\
2x - 5y + 3z &= 3 \\
x + 2y + z &= 12
\end{aligned}
\][/tex]
The equations still do not match.
- Option C:
[tex]\[
\begin{aligned}
5x + 2x + x &= 16 \\
7y - 5y + 2y &= 3 \\
-5z + 3z + z &= 12
\end{aligned}
\][/tex]
This is a nonsensical rewriting of the matrix equation and does not match the format of linear equations.
- Option D:
[tex]\[
\begin{aligned}
5x + 2y + z &= 16 \\
7x - 5y + 2z &= 3 \\
-5x + 3y + z &= 12
\end{aligned}
\][/tex]
These equations exactly match the derived equations from the matrix equation.
Therefore, the correct option is:
[tex]\[
\boxed{D}
\][/tex]
