Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l }
- 5 x + 3 y = 8 \
- 2 x + 4 y = 20
\end{array}$$
A) $\{ ( 2,6 ) \}$
B) $\{ ( - 2 , - 6 ) \}$
C) $\{ ( 6,2 ) \}$
D) $\{ ( - 6 , - 2 ) \}$

Question

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
- 5 x + 3 y = 8 \
- 2 x + 4 y = 20
\end{array}$$
A) $\{ ( 2,6 ) \}$
B) $\{ ( - 2 , - 6 ) \}$
C) $\{ ( 6,2 ) \}$
D) $\{ ( - 6 , - 2 ) \}$

Quizplus · Accepted Answer

To solve the system using the inverse of the coefficient matrix, first, we write the system in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants. For this system, $$A = \begin{pmatrix} -5 & 3 \ -2 & 4 \end{pmatrix}, X = \begin{pmatrix} x \ y \end{pmatrix}, \text{ and } B = \begin{pmatrix} 8 \ 20 \end{pmatrix}.$$ The inverse of A, $A^{-1}$, is calculated as $\frac{1}{\text{det}(A)} \begin{pmatrix} 4 & -3 \ 2 & -5 \end{pmatrix}$, where det(A) = (-5)(4) - (3)(-2) = -20 + 6 = -14. Thus, $A^{-1} = \frac{1}{-14} \begin{pmatrix} 4 & -3 \ 2 & -5 \end{pmatrix} = \begin{pmatrix} -2/7 & 3/14 \ -1/7 & 5/14 \end{pmatrix}$. Multiplying $A^{-1}$ by B gives X, $$X = A^{-1}B = \begin{pmatrix} -2/7 & 3/14 \ -1/7 & 5/14 \end{pmatrix} \begin{pmatrix} 8 \ 20 \end{pmatrix} = \begin{pmatrix} 2 \ 6 \end{pmatrix}.$$ Therefore, the solution is $x = 2, y = 6$, which corresponds to choice A.

Solve the System by Using the Inverse of the Coefficient −5x+3y=8−2x+4y=20\begin{array} { l } - 5 x + 3 y = 8 \\- 2 x + 4 y = 20\end{array}−5x+3y=8−2x+4y=20​

Solve the System by Using the Inverse of the Coefficient $\begin{array} { l } - 5 x + 3 y = 8 \\- 2 x + 4 y = 20\end{array}$