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Sagot :
Sure, let's solve the given system of linear equations step by step.
The given system of equations is:
[tex]\[ \left\{ \begin{array}{c} x + 3y + 2z = 26 \\ x - 3y + 4z = 2 \\ 2x + y + z = 8 \end{array} \right. \][/tex]
We need to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] that satisfy all three equations simultaneously.
### Step 1: Write the system as a matrix
We can represent this system as a matrix equation [tex]\(AX = B\)[/tex]:
[tex]\[ A = \begin{bmatrix} 1 & 3 & 2 \\ 1 & -3 & 4 \\ 2 & 1 & 1 \end{bmatrix} , \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} , \quad B = \begin{bmatrix} 26 \\ 2 \\ 8 \end{bmatrix} \][/tex]
### Step 2: Solve the matrix equation [tex]\(AX = B\)[/tex]
To solve for [tex]\(X\)[/tex], we typically use the inverse of matrix [tex]\(A\)[/tex] if it exists, or other methods like row reduction or computational tools.
In this case, the solution to the system has been found and is:
[tex]\[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -1.42857143 \\ 5.71428571 \\ 5.14285714 \end{bmatrix} \][/tex]
### Step 3: Convert the solution to fractions
We now convert the decimal solutions to fractions to match the provided answer choices.
[tex]\[ x = -1.42857143 = -\frac{10}{7} \][/tex]
[tex]\[ y = 5.71428571 = \frac{40}{7} \][/tex]
[tex]\[ z = 5.14285714 = \frac{36}{7} \][/tex]
### Step 4: Select the correct answer from the provided choices
The calculated solution in fractional form is:
[tex]\[ \left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right) \][/tex]
Comparing this with the given answer choices:
A. [tex]\(\left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
B. [tex]\(\left( -\frac{10}{7}, \frac{24}{7}, \frac{40}{7} \right)\)[/tex]
C. [tex]\(\left( -\frac{5}{7}, \frac{24}{7}, \frac{40}{7} \right)\)[/tex]
D. [tex]\(\left( -\frac{5}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
The correct answer is:
A. [tex]\(\left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
The given system of equations is:
[tex]\[ \left\{ \begin{array}{c} x + 3y + 2z = 26 \\ x - 3y + 4z = 2 \\ 2x + y + z = 8 \end{array} \right. \][/tex]
We need to find the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] that satisfy all three equations simultaneously.
### Step 1: Write the system as a matrix
We can represent this system as a matrix equation [tex]\(AX = B\)[/tex]:
[tex]\[ A = \begin{bmatrix} 1 & 3 & 2 \\ 1 & -3 & 4 \\ 2 & 1 & 1 \end{bmatrix} , \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} , \quad B = \begin{bmatrix} 26 \\ 2 \\ 8 \end{bmatrix} \][/tex]
### Step 2: Solve the matrix equation [tex]\(AX = B\)[/tex]
To solve for [tex]\(X\)[/tex], we typically use the inverse of matrix [tex]\(A\)[/tex] if it exists, or other methods like row reduction or computational tools.
In this case, the solution to the system has been found and is:
[tex]\[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -1.42857143 \\ 5.71428571 \\ 5.14285714 \end{bmatrix} \][/tex]
### Step 3: Convert the solution to fractions
We now convert the decimal solutions to fractions to match the provided answer choices.
[tex]\[ x = -1.42857143 = -\frac{10}{7} \][/tex]
[tex]\[ y = 5.71428571 = \frac{40}{7} \][/tex]
[tex]\[ z = 5.14285714 = \frac{36}{7} \][/tex]
### Step 4: Select the correct answer from the provided choices
The calculated solution in fractional form is:
[tex]\[ \left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right) \][/tex]
Comparing this with the given answer choices:
A. [tex]\(\left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
B. [tex]\(\left( -\frac{10}{7}, \frac{24}{7}, \frac{40}{7} \right)\)[/tex]
C. [tex]\(\left( -\frac{5}{7}, \frac{24}{7}, \frac{40}{7} \right)\)[/tex]
D. [tex]\(\left( -\frac{5}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
The correct answer is:
A. [tex]\(\left( -\frac{10}{7}, \frac{40}{7}, \frac{36}{7} \right)\)[/tex]
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