Find a solution of the system of equations and verify your result using matrix multiplication.
$$\left\{ \begin{aligned}
- 8 x + 3 y & = 20 \
3 x - 5 y & = - 9
\end{aligned} \right.$$

Question

Quizplus · Accepted Answer

We can solve this system of equations by representing the coefficients in matrix form and then using matrix inversion to solve for the variables.
First, we can represent the system of equations as the matrix equation $Ax=b$, where

$$A=\begin{bmatrix} -8 & 3 \ 3 & -5 \end{bmatrix}, \quad x=\begin{bmatrix} x \ y \end{bmatrix}, \quad 	ext{and} \quad b=\begin{bmatrix} 20 \ -9 \end{bmatrix}.$$

To solve for $x$, we can multiply both sides of the equation by $A^{-1}$, the inverse of $A$:

$$A^{-1}Ax=A^{-1}b \quad \Rightarrow \quad x=A^{-1}b.$$

Using a calculator or other method, we can find that the inverse of $A$ is

$$A^{-1}=\frac{1}{38}\begin{bmatrix} -5 & -3 \ -3 & -8 \end{bmatrix}.$$

Multiplying $A^{-1}$ by $b$, we get

$$A^{-1}b=\frac{1}{38}\begin{bmatrix} -5 & -3 \ -3 & -8 \end{bmatrix}\begin{bmatrix} 20 \ -9 \end{bmatrix}=\begin{bmatrix} -\frac{73}{31} \ \frac{12}{31} \end{bmatrix}.$$

Therefore, the solution to the system of equations is $x=\begin{bmatrix} -\frac{73}{31} \ \frac{12}{31} \end{bmatrix}$. To verify our solution, we can multiply $A$ by $x$ and check that the result is equal to $b$:

$$Ax=\begin{bmatrix} -8 & 3 \ 3 & -5 \end{bmatrix}\begin{bmatrix} -\frac{73}{31} \ \frac{12}{31} \end{bmatrix}=\begin{bmatrix} 20 \ -9 \end{bmatrix}=b.$$

Since the matrix multiplication checks out, we can be confident that our solution is correct. Therefore, the answer is D.

Find a Solution of the System of Equations and Verify {−8x+3y=203x−5y=−9\left\{ \begin{aligned}- 8 x + 3 y & = 20 \\3 x - 5 y & = - 9\end{aligned} \right.{−8x+3y3x−5y​=20=−9​

Find a Solution of the System of Equations and Verify $\left\{ \begin{aligned}- 8 x + 3 y & = 20 \\3 x - 5 y & = - 9\end{aligned} \right.$