To find the value of [tex]\(a - b + c + d\)[/tex] based on the given system of equations:
[tex]\[
\begin{cases}
4x + 2y = 7 \\
5x - 6y = 9
\end{cases}
\][/tex]
we need to first express the system in the form [tex]\( AX = C \)[/tex], where [tex]\( A \)[/tex] is the matrix of coefficients, [tex]\( X \)[/tex] is the column vector of the variables, and [tex]\( C \)[/tex] is the column vector of constants.
Examining the coefficients of the variables in the system of equations, we can construct matrix [tex]\( A \)[/tex] as:
[tex]\[
A = \begin{pmatrix}
a & c \\
b & d
\end{pmatrix} = \begin{pmatrix}
4 & 2 \\
5 & -6
\end{pmatrix}
\][/tex]
Here, from matrix [tex]\( A \)[/tex]:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = 5\)[/tex]
- [tex]\(c = 2\)[/tex]
- [tex]\(d = -6\)[/tex]
We need to calculate [tex]\(a - b + c + d \)[/tex]:
[tex]\[
a - b + c + d = 4 - 5 + 2 - 6
\][/tex]
Now, performing the arithmetic step-by-step:
1. [tex]\(4 - 5 = -1\)[/tex]
2. [tex]\(-1 + 2 = 1\)[/tex]
3. [tex]\(1 - 6 = -5\)[/tex]
Thus, the value of [tex]\(a - b + c + d\)[/tex] is:
[tex]\[
\boxed{-5}
\][/tex]
