Sure, I'll guide you through solving for [tex]\( x \)[/tex] in the given system of linear equations using Cramer's Rule step by step.
### Step 1: Identify the Coefficients and Constants
First, let's identify the coefficients of [tex]\( x, y, z \)[/tex] and the constants on the right side of the equations:
[tex]\[
\begin{array}{l}
x - 3y + 2z = 1 \quad \text{(Equation 1)} \\
3x + 4y - 5z = -2 \quad \text{(Equation 2)} \\
4x + y + z = 7 \quad \text{(Equation 3)}
\end{array}
\][/tex]
So, we have:
[tex]\[
\begin{array}{c|ccc|c}
\text{Equation} & x & y & z & \text{Constant} \\
\hline
1 & 1 & -3 & 2 & 1 \\
2 & 3 & 4 & -5 & -2 \\
3 & 4 & 1 & 1 & 7 \\
\end{array}
\][/tex]
### Step 2: Form the Coefficient Matrix [tex]\( D \)[/tex]
The coefficient matrix [tex]\( D \)[/tex] is:
[tex]\[
D = \begin{vmatrix}
1 & -3 & 2 \\
3 & 4 & -5 \\
4 & 1 & 1
\end{vmatrix}
\][/tex]
### Step 3: Form Matrices [tex]\( D_x, D_y, \)[/tex] and [tex]\( D_z \)[/tex]
To solve for [tex]\( x \)[/tex], we need matrix [tex]\( D_x \)[/tex], which is formed by replacing the [tex]\( x \)[/tex]-column in [tex]\( D \)[/tex] with the constants:
[tex]\[
D_x = \begin{vmatrix}
1 & -3 & 2 \\
-2 & 4 & -5 \\
7 & 1 & 1
\end{vmatrix}
\][/tex]
### Step 4: Compute Determinants of [tex]\( D \)[/tex] and [tex]\( D_x \)[/tex]
The determinant of any [tex]\( 3 \times 3 \)[/tex] matrix [tex]\( \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \)[/tex] is calculated as:
[tex]\[
\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
Thus for [tex]\( D \)[/tex]:
[tex]\[
D = \begin{vmatrix}
1 & -3 & 2 \\
3 & 4 & -5 \\
4 & 1 & 1
\end{vmatrix}
= 1(4 \cdot 1 - (-5) \cdot 1) - (-3)(3 \cdot 1 - (-5) \cdot 4) + 2(3 \cdot 1 - 4 \cdot 4)
= 1(4 + 5) + 3(3 + 20) + 2(3 - 16)
= 1 \cdot 9 + 3 \cdot 23 + 2 \cdot (-13)
= 9 + 69 - 26
= 52
\][/tex]
For [tex]\( D_x \)[/tex]:
[tex]\[
D_x = \begin{vmatrix}
1 & -3 & 2 \\
-2 & 4 & -5 \\
7 & 1 & 1
\end{vmatrix}
= 1((4 \cdot 1) - (-5 \cdot 1)) - (-3)((-2 \cdot 1) - (-5 \cdot 7)) + 2((-2 \cdot 1) - (4 \cdot 7))
= 1(4 + 5) + 3(2 + 35) + 2((-2) - 28)
= 9 + 111 - 60
= 60
\][/tex]
### Step 5: Apply Cramer's Rule to Determine [tex]\( x \)[/tex]
According to Cramer's Rule:
[tex]\[
x = \frac{D_x}{D}
\][/tex]
So,
[tex]\[
x = \frac{48}{52} = \frac{12}{13} \approx 0.923
\][/tex]
Thus, the value of [tex]\( x \)[/tex] in the solution to the system of equations is approximately [tex]\( 0.923 \)[/tex].
