Use matrices to solve the system of equations. If the system has no solution, so indicate .
A) ( 8, 7.5 )
B) ( 7, 6.5 )
C) ( 7, 7.5 )
D) ( 8, 6.5 )
E) No solution.

Question

Use matrices to solve the system of equations. If the system has no solution, so indicate . 
A) ( 8, 7.5 ) 
B) ( 7, 6.5 ) 
C) ( 7, 7.5 ) 
D) ( 8, 6.5 ) 
E) No solution.

Quizplus · Accepted Answer

We can write the system of equations in matrix form as 
\begin{bmatrix} 6 & 3 \ 4 & 5 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 45 \ 47 \end{bmatrix}
We can solve for the solution vector by multiplying both sides by the inverse of the coefficient matrix. 
The coefficient matrix has an inverse since its determinant is $(6)(5)-(3)(4)=18
eq 0$. 
Calculating the inverse, we have 
$$\begin{bmatrix} 6 & 3 \ 4 & 5 \end{bmatrix}^{-1} = \frac{1}{18}\begin{bmatrix} 5 & -3 \ -4 & 6 \end{bmatrix}$$
Multiplying both sides by the inverse, we get 
\begin{align*}
\begin{bmatrix} x \ y \end{bmatrix} &= \frac{1}{18}\begin{bmatrix} 5 & -3 \ -4 & 6 \end{bmatrix} \begin{bmatrix} 45 \ 47 \end{bmatrix} \
&= \begin{bmatrix} 7 \ 7.5 \end{bmatrix}
\end{align*}
Therefore, the only solution is (7,7.5), so the answer is C.

Use Matrices to Solve the System of Equations