Solve the system using the inverse method.
-$$\left\{ \begin{aligned}
x + 2 y + 3 z & = 11 \
x + y + z & = 3 \
2 x + 2 y + z & = 10
\end{aligned} \right.$$
A) $x = - 8 , y = 49 , z = 7$
B) $x = 33 , y = 46 , z = 16$
C) $x = - 9 , y = 16 , z = - 4$
D) $x = 44 , y = - 6 , z = 10$

Question

Quizplus · Accepted Answer

To solve the system using the inverse method, we first 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 the given system:$$A = \begin{pmatrix} 1 & 2 & 3 \ 1 & 1 & 1 \ 2 & 2 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \ z \end{pmatrix}, \quad B = \begin{pmatrix} 11 \ 3 \ 10 \end{pmatrix}.$$To find $X$, we compute $X = A^{-1}B$. Calculating the inverse of $A$ and then multiplying it by $B$ gives the solution for $X$, which corresponds to the values of $x$, $y$, and $z$. After performing these calculations, the correct solution is $x = -9$, $y = 16$, and $z = -4$, which matches option C.

Solve the System Using the Inverse Method {x+2y+3z=11x+y+z=32x+2y+z=10\left\{ \begin{aligned}x + 2 y + 3 z & = 11 \\x + y + z & = 3 \\2 x + 2 y + z & = 10\end{aligned} \right.⎩⎨⎧​x+2y+3zx+y+z2x+2y+z​=11=3=10​

Solve the System Using the Inverse Method $\left\{ \begin{aligned}x + 2 y + 3 z & = 11 \\x + y + z & = 3 \\2 x + 2 y + z & = 10\end{aligned} \right.$