Use an inverse matrix to solve (if possible) the system of linear equations. $\left\{ \begin{array} { l }
0.2 x - 0.6 y = 14.4 \
- x + 1.4 y = - 20.8
\end{array} \right.$
A)(-24, -31)
B)(-23, -32)
C)(-22, -32)
D)(-24, -32)
E)(-23, -30)

Question

Use an inverse matrix to solve (if possible) the system of linear equations.  $\left\{ \begin{array} { l } 
0.2 x - 0.6 y = 14.4 \
- x + 1.4 y = - 20.8
\end{array} \right.$  
A)(-24, -31)
B)(-23, -32)
C)(-22, -32)
D)(-24, -32)
E)(-23, -30)

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 the given system:$$A = \begin{pmatrix} 0.2 & -0.6 \ -1 & 1.4 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \end{pmatrix}, \quad B = \begin{pmatrix} 14.4 \ -20.8 \end{pmatrix}$$To find the solution, we compute $X = A^{-1}B$, where $A^{-1}$ is the inverse of matrix A. The inverse of A, $A^{-1}$, can be found using the formula:$$A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$$where $a, b, c, d$ are the elements of matrix A, so in this case, $a = 0.2, b = -0.6, c = -1, d = 1.4$, and $\text{det}(A) = (0.2)(1.4) - (-0.6)(-1) = 0.28 - 0.6 = -0.32$.Thus,$$A^{-1} = \frac{1}{-0.32} \begin{pmatrix} 1.4 & 0.6 \ 1 & 0.2 \end{pmatrix} = \begin{pmatrix} -4.375 & -1.875 \ -3.125 & -0.625 \end{pmatrix}$$Multiplying $A^{-1}$ by B gives:$$X = \begin{pmatrix} -4.375 & -1.875 \ -3.125 & -0.625 \end{pmatrix} \begin{pmatrix} 14.4 \ -20.8 \end{pmatrix} = \begin{pmatrix} -24 \ -32 \end{pmatrix}$$Therefore, the solution to the system of equations is $x = -24$ and $y = -32$, which corresponds to choice D.

Use an Inverse Matrix to Solve (If Possible) the System {0.2x−0.6y=14.4−x+1.4y=−20.8\left\{ \begin{array} { l } 0.2 x - 0.6 y = 14.4 \\- x + 1.4 y = - 20.8\end{array} \right.{0.2x−0.6y=14.4−x+1.4y=−20.8​

Use an Inverse Matrix to Solve (If Possible) the System $\left\{ \begin{array} { l } 0.2 x - 0.6 y = 14.4 \\- x + 1.4 y = - 20.8\end{array} \right.$