Question 349961
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Let *[tex \Large x] represent the number of pill #1.


Let *[tex \Large y] represent the number of pill #2.


Let *[tex \Large z] represent the number of pill #3.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ \,50x\ +\ 100y\ +\ 50z\ \ \ =\ 350]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 1000x\ +\ 200y\ +\ 500z\ =\ 4200]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \,100x\ +\ 100y\ \ \ \ \ \ \ \ \ \ \ =\ 400]


Divide the first equation by 50, the second by 100, and the third by 100.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ \,x\ +\ 2y\ +\ z\ \ \ =\ 7]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 10x\ +\ 2y\ +\ 5z\ \,=\ 42]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ \,x\ +\ y\ \ \ \ \ \ \ \ \ \ \, =\ 4]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 2 & 1 & 7 \\ 10 & 2 & 5 & 42 \\ 1 & 1 & 0 & 4 \end{array}\right|]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -10R_1\ +\ R_2\ \rightarrow\ R_2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 2 & 1 & 7 \\ 0 & -18 & -5 & -28 \\ 1 & 1 & 0 & 4 \end{array}\right|]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -R_1\ +\ R_3\ \rightarrow\ R_3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 2 & 1 & 7 \\ 0 & -18 & -5 & -28 \\ 0 & -1 & -1 & -3 \end{array}\right|]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{R_2}{-18}\ \rightarrow\ R_2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 2 & 1 & 7 \\ 0 & 1 & \frac{5}{18} & \frac{14}{9} \\ 0 & -1 & -1 & -3 \end{array}\right|]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ R_2\ +\ R_3\ \rightarrow\ R_3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 2 & 1 & 7 \\ 0 & 1 & \frac{5}{18} & \frac{14}{9} \\ 0 & 0 & \frac{-13}{18} & \frac{-13}{9} \end{array}\right|]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{-18R_3}{13}\ \rightarrow\ R_3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 2 & 1 & 7 \\ 0 & 1 & \frac{5}{18} & \frac{14}{9} \\ 0 & 0 & 1 & 2 \end{array}\right|]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{-5R_3}{18}\ +\ R_2\ \rightarrow\ R_2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 2 & 1 & 7 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right|]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -2R_2\ +\ R_1\ \rightarrow\ R_1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 0 & 1 & 5 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right|]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -R_3\ +\ R_1\ \rightarrow\ R_1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left| \begin{array}{cccc}1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right|]


Solution set: *[tex \LARGE \left{(x,\,y,\,z)\ |\ (x,\,y,\,z)\ =\ (3,\,1,\,2)\right}]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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