To solve for \( a - b + c + d \), we need to gather the coefficients given in the system of equations and then perform the necessary arithmetic operations.
Here is the system of linear equations we are given:
[tex]\[
\begin{array}{l}
2x + 8y = 7 \\
4x - 2y = 9
\end{array}
\][/tex]
We can express this system in matrix form \( AX = C \), where \( A \) is the matrix of coefficients, \( X \) is the column vector of variables \( x \) and \( y \), and \( C \) is the column vector of the constants on the right-hand side of the equations.
Matrix \( A \) is defined by the coefficients of \( x \) and \( y \) in each equation:
[tex]\[
A = \begin{pmatrix}
a & c \\
b & d
\end{pmatrix}
\][/tex]
From our given equations, we can identify the coefficients:
From the first equation \( 2x + 8y = 7 \):
- The coefficient of \( x \) is \( 2 \), so \( a = 2 \)
- The coefficient of \( y \) is \( 8 \), so \( c = 8 \)
From the second equation \( 4x - 2y = 9 \):
- The coefficient of \( x \) is \( 4 \), so \( b = 4 \)
- The coefficient of \( y \) is \( -2 \), so \( d = -2 \)
We now have:
[tex]\[
a = 2, \quad b = 4, \quad c = 8, \quad d = -2
\][/tex]
Next, we calculate \( a - b + c + d \):
[tex]\[
a - b + c + d = 2 - 4 + 8 - 2
\][/tex]
Performing the arithmetic step by step:
[tex]\[
2 - 4 = -2
\][/tex]
[tex]\[
-2 + 8 = 6
\][/tex]
[tex]\[
6 - 2 = 4
\][/tex]
Thus, the value of \( a - b + c + d \) is:
[tex]\[
4
\][/tex]
