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If the following system of equations is written as a matrix equation in the form [tex]A X = C[/tex], and matrix [tex]A[/tex] is 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}
4x + 2y = 7 \\
5x - 6y = 9
\end{array}
[/tex]

Answer here:



Answer :

GinnyAnswer
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]

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