To solve the system of linear equations:
[tex]\[
\begin{cases}
-2x + 14y = 148 \\
3x + 5y = 246
\end{cases}
\][/tex]
we can use Cramer's Rule, which provides a straightforward way to solve for each variable. The general form of Cramer's Rule for a 2x2 system is:
[tex]\[
x = \frac{\Delta_x}{\Delta} \quad \text{and} \quad y = \frac{\Delta_y}{\Delta}
\][/tex]
where [tex]\(\Delta\)[/tex] is the determinant of the coefficient matrix, [tex]\(\Delta_x\)[/tex] is the determinant of the matrix formed by replacing the [tex]\(x\)[/tex]-column with the constants from the right-hand side, and [tex]\(\Delta_y\)[/tex] is the determinant of the matrix formed by replacing the [tex]\(y\)[/tex]-column with the constants from the right-hand side.
Let's define the matrices involved:
1. The coefficient matrix [tex]\(A\)[/tex]:
[tex]\[
A = \begin{pmatrix}
-2 & 14 \\
3 & 5
\end{pmatrix}
\][/tex]
2. The determinant [tex]\(\Delta\)[/tex] of matrix [tex]\(A\)[/tex]:
[tex]\[
\Delta = \left| \begin{array}{cc}
-2 & 14 \\
3 & 5
\end{array} \right|
= (-2 \cdot 5) - (14 \cdot 3)
= -10 - 42
= -52
\][/tex]
3. The matrix [tex]\(B_1\)[/tex] to solve for [tex]\(x\)[/tex], replacing the first column of [tex]\(A\)[/tex] with the constants:
[tex]\[
B_1 = \begin{pmatrix}
148 & 14 \\
246 & 5
\end{pmatrix}
\][/tex]
The determinant [tex]\(\Delta_x\)[/tex]:
[tex]\[
\Delta_x = \left| \begin{array}{cc}
148 & 14 \\
246 & 5
\end{array} \right|
= (148 \cdot 5) - (14 \cdot 246)
= 740 - 3444
= -2704
\][/tex]
4. The matrix [tex]\(B_2\)[/tex] to solve for [tex]\(y\)[/tex], replacing the second column of [tex]\(A\)[/tex] with the constants:
[tex]\[
B_2 = \begin{pmatrix}
-2 & 148 \\
3 & 246
\end{pmatrix}
\][/tex]
The determinant [tex]\(\Delta_y\)[/tex]:
[tex]\[
\Delta_y = \left| \begin{array}{cc}
-2 & 148 \\
3 & 246
\end{array} \right|
= (-2 \cdot 246) - (148 \cdot 3)
= -492 - 444
= -936
\][/tex]
Now, apply Cramer's Rule to find [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
x = \frac{\Delta_x}{\Delta} = \frac{-2704}{-52} = 52
\][/tex]
[tex]\[
y = \frac{\Delta_y}{\Delta} = \frac{-936}{-52} = 18
\][/tex]
Thus, we can determine the [tex]\(x\)[/tex]-value of the solution to the system of linear equations using the equation:
[tex]\[
x = \frac{\left| \begin{array}{cc}
148 & 14 \\
246 & 5
\end{array} \right|}{\left| \begin{array}{cc}
-2 & 14 \\
3 & 5
\end{array} \right|} = 52
\][/tex]
Therefore, the equation that can be used to determine the [tex]\(x\)[/tex]-value of the solution is:
[tex]\[
\frac{\left| \begin{array}{cc}
148 & 14 \\
246 & 5
\end{array} \right|}{\left| \begin{array}{cc}
-2 & 14 \\
3 & 5
\end{array} \right|} = 52
\][/tex]
