To solve the problem, we need to work with the given system of linear equations:
[tex]\[
\begin{align*}
4x + 2y &= 7 \\
5x - 6y &= 9
\end{align*}
\][/tex]
First, we write these equations in matrix form [tex]\(AX = C\)[/tex], where [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(X\)[/tex] is the column vector of variables, and [tex]\(C\)[/tex] is the constant matrix.
The system of equations can be written as:
[tex]\[
\begin{pmatrix}
4 & 2 \\
5 & -6
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
7 \\
9
\end{pmatrix}
\][/tex]
Here we have:
[tex]\[ A = \begin{pmatrix}
a & c \\
b & d
\end{pmatrix} \][/tex]
Therefore, comparing it with the coefficient matrix from the equations, we identify:
[tex]\[ a = 4, \; c = 2, \; b = 5, \; d = -6 \][/tex]
Now, to find the value of [tex]\(a - b + c + d\)[/tex]:
Let’s substitute the known values into the expression [tex]\(a - b + c + d\)[/tex]:
[tex]\[
a - b + c + d = 4 - 5 + 2 + (-6)
\][/tex]
Performing the arithmetic step-by-step:
[tex]\[
a - b = 4 - 5 = -1
\][/tex]
[tex]\[
-1 + c = -1 + 2 = 1
\][/tex]
[tex]\[
1 + d = 1 + (-6) = -5
\][/tex]
Thus, the final result is:
[tex]\[
a - b + c + d = -5
\][/tex]
So, the value of [tex]\(a - b + c + d\)[/tex] is [tex]\(\boxed{-5}\)[/tex].
