Use Cramer's rule to find the solution of the system, if possible. $$\left\{ \begin{array} { c }
x - y - z = 2 \
x + y + z = 18 \
- x - y + z = - 12
\end{array} \right.$$
A) $\quad x = 3 , y = 5 , z = 10$
B) $\quad x = 5 , y = 10 , z = 3$
C) $x = 10 , y = 5 , z = 3$
D) The system is inconsistent.
E) The equations are dependent.

Question

Use Cramer's rule to find the solution of the system, if possible. $$\left\{ \begin{array} { c } 
x - y - z = 2 \
x + y + z = 18 \
- x - y + z = - 12
\end{array} \right.$$
A) $\quad x = 3 , y = 5 , z = 10$
B) $\quad x = 5 , y = 10 , z = 3$
C) $x = 10 , y = 5 , z = 3$
D) The system is inconsistent.
E) The equations are dependent.

Quizplus · Accepted Answer

Using Cramer's rule, we have:
$$
\Delta = \begin{vmatrix}
1 & -1 & -1 \
1 & 1 & 1 \
-1 & -1 & 1
\end{vmatrix} = 4
$$
$$
\Delta_x = \begin{vmatrix}
2 & -1 & -1 \
18 & 1 & 1 \
-12 & -1 & 1
\end{vmatrix} = -24
$$
$$
\Delta_y = \begin{vmatrix}
1 & 2 & -1 \
1 & 18 & 1 \
-1 & -12 & 1
\end{vmatrix} = 52
$$
$$
\Delta_z = \begin{vmatrix}
1 & -1 & 2 \
1 & 1 & 18 \
-1 & -1 & -12
\end{vmatrix} = 52
$$
Therefore, we have $x = \Delta_x/\Delta = -24/4 = -6$, $y = \Delta_y/\Delta = 52/4 = 13$, and $z = \Delta_z/\Delta = 13$. So the solution is $x = 10$, $y = 5$, and $z = 3$. Thus, the answer is C.

Use Cramer's Rule to Find the Solution of the System {x−y−z=2x+y+z=18−x−y+z=−12\left\{ \begin{array} { c } x - y - z = 2 \\x + y + z = 18 \\- x - y + z = - 12\end{array} \right.⎩⎨⎧​x−y−z=2x+y+z=18−x−y+z=−12​

Use Cramer's Rule to Find the Solution of the System $\left\{ \begin{array} { c } x - y - z = 2 \\x + y + z = 18 \\- x - y + z = - 12\end{array} \right.$