At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

If the following system of equations were written as a matrix equation in the form [tex]$A X = C$[/tex], and matrix [tex]$A$[/tex] were expressed in the form [tex]$A = \left[\begin{array}{ll}a & c \\ b & d\end{array}\right]$[/tex], find the value of [tex]$a - b + c + d$[/tex].

[tex]\[
\begin{array}{l}
2x + 8y = 7 \\
4x - 2y = 9
\end{array}
\][/tex]


Sagot :

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]