Solve: \( \left [ \begin{array} { c c c } { 3 } & { - 1 } & { - 2 } \\ { 9 } & { - 4 } & { - 8 } \\ { - 3 } & { 2 } & { 6 } \end{array} \right ] \vec { x } = \left [ \begin{array} { c } { - 21 } \\ { - 78 } \\ { 48 } \end{array} \right ] \) 

\( \vec { x } = \) 

Write out the matrix equation for this given system of equations 

\( \left \{ \begin{array} { r } { - 8 r + 11 z + 4 y = 300 } \\ { - 3 r + 4 z + 2 y = 250 } \\ { y = 100 } \end{array} \right . \) 

Give the solution as an ordered triple: \( ( r , z , y ) = \square \) 

Find the soltuion to the system \( \left \{ \begin{array} { l } { a _ { 1 } x + b _ { 1 } y + c _ { 1 } z = - 2 } \\ { a _ { 2 } x + b _ { 2 } y + c _ { 2 } z = - 3 } \\ { a _ { 3 } x + b _ { 3 } y + c _ { 3 } z = - 2 } \end{array} \right . \) 

given the inverse for the martix \( \left [ \begin{array} { l } { a _ { 1 } b _ { 1 } c _ { 1 } } \\ { a _ { 2 } b _ { 2 } c _ { 2 } } \\ { a _ { 3 } b _ { 3 } c _ { 3 } } \end{array} \right ] \) is 

\( \left [ \begin{array} { c c c } { - 3 } & { - 3 } & { 7 } \\ { - 1 } & { - 1 } & { 2 } \\ { 2 } & { 1 } & { - 5 } \end{array} \right ] \) 

\(\left [ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right ] = \)

Find the soltulon to the system \( \left \{ \begin{array} { l } { a _ { 1 } x + b _ { 1 } y + c _ { 1 } z = - 11 } \\ { a _ { 2 } x + b _ { 2 } y + c _ { 2 } z = 1 } \\ { a _ { 3 } x + b _ { 3 } y + c _ { 3 } z = - 4 } \end{array} \right . \) 

given the inverse for the martix \( \left [ \begin{array} { l } { a _ { 1 } b _ { 1 } c _ { 1 } } \\ { a _ { 2 } b _ { 2 } c _ { 2 } } \\ { a _ { 3 } b _ { 3 } c _ { 3 } } \end{array} \right ] \) is 

\( \left [ \begin{array} { c c c } { - 3 } & { - 3 } & { 7 } \\ { - 1 } & { - 1 } & { 2 } \\ { 2 } & { 1 } & { - 5 } \end{array} \right ] \) 

\(\left [ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right ] = \) 

An exponential decay function can be used to model the number of grams of a radioactive material that remain after a period of time. A radioactive isotope decays over time, with the amount remaining after \( t \) years given by 

\( y = 150 e ^ { - 0.00012608 t } \) if 150 grams is the original amount. 

(a) How much remains after 1000 years? 

(b) Use graphical methods to estimate the number of years until 30 grams of a radioactive isotope remain. 

(a) The amount of a radioactive isotope remaining after 1000 years is approximately \( \square \) grams. 

(Type an integer or decimal rounded to the nearest tenth as needed.) 

(b) What is the number of years until 30 grams of a radioactive isotope remain? Choose the correct answer below. 

A. about 13,765 years                 B. about 11,765 years   

C. about 6383 years                 D. about 12,765 years