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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]
[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]
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