Certainly! Let's break this problem down step-by-step.
Given the system of equations:
[tex]\[
\begin{cases}
2x + 8y = 7 \\
4x - 2y = 9
\end{cases}
\][/tex]
We want to represent these equations in the form [tex]\( A X = C \)[/tex]:
- [tex]\( A \)[/tex] is the matrix containing the coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- [tex]\( X \)[/tex] is the column matrix of the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- [tex]\( C \)[/tex] is the column matrix of the constants on the right hand side of the equations.
The system can be written as:
[tex]\[
\begin{pmatrix}
2 & 8 \\
4 & -2
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix} =
\begin{pmatrix}
7 \\
9
\end{pmatrix}
\][/tex]
From this, we identify:
[tex]\[
A = \begin{pmatrix}
a & c \\
b & d
\end{pmatrix} =
\begin{pmatrix}
2 & 8 \\
4 & -2
\end{pmatrix}
\][/tex]
Now we can extract the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] directly:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 8 \)[/tex]
- [tex]\( d = -2 \)[/tex]
Next, we need to find the value of [tex]\( a - b + c + d \)[/tex]:
[tex]\[
a - b + c + d = 2 - 4 + 8 - 2
\][/tex]
Calculating step by step:
[tex]\[
2 - 4 = -2
\][/tex]
[tex]\[
-2 + 8 = 6
\][/tex]
[tex]\[
6 - 2 = 4
\][/tex]
Thus, the value of [tex]\( a - b + c + d \)[/tex] is:
[tex]\[
4
\][/tex]
Therefore, the final result is:
[tex]\[
a - b + c + d = 4
\][/tex]
