If the following system of equations was written as a matrix equation in the form AX = C, and matrix A was expressed in the form: A= {A C} {B D}, find the value of a-b +c+d. 2x+8y=7 4x-2y=9 Please help, i dont know which number would be which letters
The Correct Answer and Explanation is:
We are given the system of equations:
[
2x + 8y = 7
]
[
4x – 2y = 9
]
We are asked to express this system as a matrix equation of the form ( AX = C ), where matrix ( A ) is given by:
[
A = \begin{pmatrix} A & C \ B & D \end{pmatrix}
]
Here, we need to match the values of ( A ), ( B ), ( C ), and ( D ) from the system of equations, and then find the value of ( a – b + c + d ).
Step 1: Write the system as a matrix equation
The system of equations can be written as:
[
\begin{pmatrix} 2 & 8 \ 4 & -2 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 7 \ 9 \end{pmatrix}
]
This corresponds to the general matrix equation ( AX = C ), where:
- Matrix ( A ) is the coefficient matrix of the variables ( x ) and ( y ), which is:
[
A = \begin{pmatrix} 2 & 8 \ 4 & -2 \end{pmatrix}
]
- Matrix ( X ) is the column matrix representing the variables:
[
X = \begin{pmatrix} x \ y \end{pmatrix}
]
- Matrix ( C ) is the column matrix containing the constants on the right-hand side of the equations:
[
C = \begin{pmatrix} 7 \ 9 \end{pmatrix}
]
Step 2: Assign the values to ( A ), ( B ), ( C ), and ( D )
We are asked to express matrix ( A ) in the form:
[
A = \begin{pmatrix} A & C \ B & D \end{pmatrix}
]
By comparing the two matrices, we see that:
- ( A = 2 ) (the top-left entry of the matrix)
- ( C = 8 ) (the top-right entry of the matrix)
- ( B = 4 ) (the bottom-left entry of the matrix)
- ( D = -2 ) (the bottom-right entry of the matrix)
Step 3: Calculate ( a – b + c + d )
Now, we need to calculate:
[
a – b + c + d
]
Substitute the values:
[
a – b + c + d = 2 – 4 + 8 – 2
]
Simplify the expression:
[
2 – 4 + 8 – 2 = 4
]
Final Answer
The value of ( a – b + c + d ) is ( \boxed{4} ).