Solve the system of equations by using an inverse matrix.
-$$\begin{array} { c }
5 x + 4 y = 8 \
4 x - 2 y = 22
\end{array}$$
A) $( 4 , - 3 )$
B) $( - 4,3 )$
C) $( 3 , - 4 )$
D) $( - 3,4 )$

Question

Solve the system of equations by using an inverse matrix.
-$$\begin{array} { c } 
5 x + 4 y = 8 \
4 x - 2 y = 22
\end{array}$$
A) $( 4 , - 3 )$
B) $( - 4,3 )$
C) $( 3 , - 4 )$
D) $( - 3,4 )$

Quizplus · Accepted Answer

To solve the system of equations using an inverse matrix, we first write the system in matrix form as AX = B, where A is the matrix of coefficients, X is the column matrix of variables, and B is the column matrix of constants. For this system, $$A = \begin{pmatrix} 5 & 4 \ 4 & -2 \end{pmatrix}, X = \begin{pmatrix} x \ y \end{pmatrix}, B = \begin{pmatrix} 8 \ 22 \end{pmatrix}$$. The solution is given by $X = A^{-1}B$. The inverse of A, $A^{-1}$, is $\frac{1}{(5)(-2) - (4)(4)} \begin{pmatrix} -2 & -4 \ -4 & 5 \end{pmatrix} = \frac{1}{-10 - 16} \begin{pmatrix} -2 & -4 \ -4 & 5 \end{pmatrix} = \frac{1}{-26} \begin{pmatrix} -2 & -4 \ -4 & 5 \end{pmatrix}$. Multiplying $A^{-1}$ by B gives $X = \frac{1}{-26} \begin{pmatrix} -2 & -4 \ -4 & 5 \end{pmatrix} \begin{pmatrix} 8 \ 22 \end{pmatrix} = \frac{1}{-26} \begin{pmatrix} -2(8) - 4(22) \ -4(8) + 5(22) \end{pmatrix} = \frac{1}{-26} \begin{pmatrix} -104 \ 70 \end{pmatrix} = \begin{pmatrix} 4 \ -3 \end{pmatrix}$. Therefore, the solution is $x = 4, y = -3$, which corresponds to choice A.

Solve the System of Equations by Using an Inverse Matrix 5x+4y=84x−2y=22\begin{array} { c } 5 x + 4 y = 8 \\4 x - 2 y = 22\end{array}5x+4y=84x−2y=22​

Solve the System of Equations by Using an Inverse Matrix $\begin{array} { c } 5 x + 4 y = 8 \\4 x - 2 y = 22\end{array}$