To solve the given system of equations, we will use the method of determinants known as Cramer's Rule. The system of equations is:
[tex]\[
\begin{array}{l}
2x + 7y = 4 \\
-4x - 3y = 14
\end{array}
\][/tex]
First, we need to determine the determinant of the coefficient matrix. The coefficient matrix is:
[tex]\[
\begin{pmatrix}
2 & 7 \\
-4 & -3
\end{pmatrix}
\][/tex]
The determinant [tex]\(\Delta\)[/tex] of this matrix is calculated as follows:
[tex]\[
\Delta = 2 \cdot (-3) - (-4) \cdot 7 = -6 + 28 = 22
\][/tex]
Next, we need to find the determinants [tex]\(\Delta_x\)[/tex] and [tex]\(\Delta_y\)[/tex], which will help us find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
1. The determinant [tex]\(\Delta_x\)[/tex] is found by replacing the first column of the coefficient matrix with the constants from the right-hand side of the equations:
[tex]\[
\Delta_x =
\begin{vmatrix}
4 & 7 \\
14 & -3
\end{vmatrix}
\][/tex]
[tex]\[
\Delta_x = 4 \cdot (-3) - 7 \cdot 14 = -12 - 98 = -110
\][/tex]
2. The determinant [tex]\(\Delta_y\)[/tex] is found by replacing the second column of the coefficient matrix with the constants from the right-hand side of the equations:
[tex]\[
\Delta_y =
\begin{vmatrix}
2 & 4 \\
-4 & 14
\end{vmatrix}
\][/tex]
[tex]\[
\Delta_y = 2 \cdot 14 - 4 \cdot (-4) = 28 + 16 = 44
\][/tex]
Now, we can find [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using the formulas:
[tex]\[
x = \frac{\Delta_x}{\Delta} = \frac{-110}{22} = -5
\][/tex]
[tex]\[
y = \frac{\Delta_y}{\Delta} = \frac{44}{22} = 2
\][/tex]
Therefore, the solution to the system of equations is the ordered pair:
[tex]\[
(x, y) = (-5, 2)
\][/tex]
