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Which system of equations can be solved using this matrix equation?

[tex]\[
\left[\begin{array}{ccc}
5 & 2 & 1 \\
7 & -5 & 2 \\
-5 & 3 & 1
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{c}
16 \\
3 \\
12
\end{array}\right]
\][/tex]

A.
[tex]\[
\begin{aligned}
5 x+7 y-5 z & =12 \\
2 x-5 y+3 z & =3 \\
x+2 y+z & =16
\end{aligned}
\][/tex]

B.
[tex]\[
\begin{aligned}
5 x+7 y-5 z & =16 \\
2 x-5 y+3 z & =3 \\
x+2 y+z & =12
\end{aligned}
\][/tex]

C.
[tex]\[
\begin{aligned}
5 x+2 x+x & =16 \\
7 y-5 y+2 y & =3 \\
-5 z+3 z+z & =12
\end{aligned}
\][/tex]

D.
[tex]\[
\begin{aligned}
5 x+2 y+z & =16 \\
7 x-5 y+2 z & =3 \\
-5 x+3 y+z & =12
\end{aligned}
\][/tex]


Sagot :

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]