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Sagot :
Certainly! Let's solve the given system of linear equations using matrix methods step-by-step.
Given system of equations can be represented as:
[tex]\[ \begin{bmatrix} 2 & 7 \\ 2 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 6 \end{bmatrix} \][/tex]
This can be written as [tex]\( \mathbf{A} \mathbf{X} = \mathbf{B} \)[/tex], where:
[tex]\[ \mathbf{A} = \begin{bmatrix} 2 & 7 \\ 2 & 6 \end{bmatrix}, \quad \mathbf{X} = \begin{bmatrix} x \\ y \end{bmatrix}, \quad \text{and} \quad \mathbf{B} = \begin{bmatrix} 8 \\ 6 \end{bmatrix} \][/tex]
We need to find the vector [tex]\( \mathbf{X} \)[/tex], which contains the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. To do this, we use the formula:
[tex]\[ \mathbf{X} = \mathbf{A}^{-1} \mathbf{B} \][/tex]
where [tex]\( \mathbf{A}^{-1} \)[/tex] is the inverse of the matrix [tex]\( \mathbf{A} \)[/tex].
### Step 1: Find the Inverse of Matrix [tex]\(\mathbf{A}\)[/tex]
Matrix [tex]\(\mathbf{A}\)[/tex] is:
[tex]\[ \mathbf{A} = \begin{bmatrix} 2 & 7 \\ 2 & 6 \end{bmatrix} \][/tex]
The inverse of a 2x2 matrix [tex]\(\mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)[/tex] is given by:
[tex]\[ \mathbf{A}^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \][/tex]
For our matrix, [tex]\( a = 2 \)[/tex], [tex]\( b = 7 \)[/tex], [tex]\( c = 2 \)[/tex], and [tex]\( d = 6 \)[/tex].
Calculating the determinant ([tex]\( ad - bc \)[/tex]):
[tex]\[ ad - bc = (2 \cdot 6) - (7 \cdot 2) = 12 - 14 = -2 \][/tex]
Substitute these values into the formula for the inverse:
[tex]\[ \mathbf{A}^{-1} = \frac{1}{-2} \begin{bmatrix} 6 & -7 \\ -2 & 2 \end{bmatrix} = \begin{bmatrix} -\frac{6}{2} & \frac{7}{2} \\ \frac{2}{2} & -\frac{2}{2} \end{bmatrix} = \begin{bmatrix} -3 & 3.5 \\ 1 & -1 \end{bmatrix} \][/tex]
### Step 2: Multiply the Inverse of Matrix [tex]\(\mathbf{A}\)[/tex] with Matrix [tex]\(\mathbf{B}\)[/tex]
[tex]\[ \mathbf{X} = \mathbf{A}^{-1} \mathbf{B} = \begin{bmatrix} -3 & 3.5 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 8 \\ 6 \end{bmatrix} \][/tex]
Perform the matrix multiplication:
For [tex]\( x \)[/tex]:
[tex]\[ x = (-3 \cdot 8) + (3.5 \cdot 6) = -24 + 21 = -3 \][/tex]
For [tex]\( y \)[/tex]:
[tex]\[ y = (1 \cdot 8) + (-1 \cdot 6) = 8 - 6 = 2 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ x = -3, \quad y = 2 \][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the system of equations are [tex]\( x = -3 \)[/tex] and [tex]\( y = 2 \)[/tex].
Given system of equations can be represented as:
[tex]\[ \begin{bmatrix} 2 & 7 \\ 2 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 6 \end{bmatrix} \][/tex]
This can be written as [tex]\( \mathbf{A} \mathbf{X} = \mathbf{B} \)[/tex], where:
[tex]\[ \mathbf{A} = \begin{bmatrix} 2 & 7 \\ 2 & 6 \end{bmatrix}, \quad \mathbf{X} = \begin{bmatrix} x \\ y \end{bmatrix}, \quad \text{and} \quad \mathbf{B} = \begin{bmatrix} 8 \\ 6 \end{bmatrix} \][/tex]
We need to find the vector [tex]\( \mathbf{X} \)[/tex], which contains the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. To do this, we use the formula:
[tex]\[ \mathbf{X} = \mathbf{A}^{-1} \mathbf{B} \][/tex]
where [tex]\( \mathbf{A}^{-1} \)[/tex] is the inverse of the matrix [tex]\( \mathbf{A} \)[/tex].
### Step 1: Find the Inverse of Matrix [tex]\(\mathbf{A}\)[/tex]
Matrix [tex]\(\mathbf{A}\)[/tex] is:
[tex]\[ \mathbf{A} = \begin{bmatrix} 2 & 7 \\ 2 & 6 \end{bmatrix} \][/tex]
The inverse of a 2x2 matrix [tex]\(\mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)[/tex] is given by:
[tex]\[ \mathbf{A}^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \][/tex]
For our matrix, [tex]\( a = 2 \)[/tex], [tex]\( b = 7 \)[/tex], [tex]\( c = 2 \)[/tex], and [tex]\( d = 6 \)[/tex].
Calculating the determinant ([tex]\( ad - bc \)[/tex]):
[tex]\[ ad - bc = (2 \cdot 6) - (7 \cdot 2) = 12 - 14 = -2 \][/tex]
Substitute these values into the formula for the inverse:
[tex]\[ \mathbf{A}^{-1} = \frac{1}{-2} \begin{bmatrix} 6 & -7 \\ -2 & 2 \end{bmatrix} = \begin{bmatrix} -\frac{6}{2} & \frac{7}{2} \\ \frac{2}{2} & -\frac{2}{2} \end{bmatrix} = \begin{bmatrix} -3 & 3.5 \\ 1 & -1 \end{bmatrix} \][/tex]
### Step 2: Multiply the Inverse of Matrix [tex]\(\mathbf{A}\)[/tex] with Matrix [tex]\(\mathbf{B}\)[/tex]
[tex]\[ \mathbf{X} = \mathbf{A}^{-1} \mathbf{B} = \begin{bmatrix} -3 & 3.5 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} 8 \\ 6 \end{bmatrix} \][/tex]
Perform the matrix multiplication:
For [tex]\( x \)[/tex]:
[tex]\[ x = (-3 \cdot 8) + (3.5 \cdot 6) = -24 + 21 = -3 \][/tex]
For [tex]\( y \)[/tex]:
[tex]\[ y = (1 \cdot 8) + (-1 \cdot 6) = 8 - 6 = 2 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ x = -3, \quad y = 2 \][/tex]
Therefore, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the system of equations are [tex]\( x = -3 \)[/tex] and [tex]\( y = 2 \)[/tex].
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