Problem $1$

A particular chemical substance is produced from three different ingredients $A,B,$ and $C,$ each of which have be dissolved in

water first before they react to form the desired substance. Suppose that a solution containing ingredient $A$ at a concentration

of $3\frac{g}{c}m3$ is combined with a solution containing ingredient $B$ at a concentration of $4\mathrm{.}2\frac{g}{c}m3$ and with a solution containing

ingredient $C$ at a concentration of $6\mathrm{.}1\frac{g}{c}m3$ to form $22\mathrm{.}4g$ of the substance. If the concentrations of $A,B,$ and $C$ in these

solutions are changed to $5\mathrm{.}4\frac{g}{c}m3,4\mathrm{.}8\frac{g}{c}m3,$ and $5\mathrm{.}5\frac{g}{c}m3,$ respectively $($while the volumes remain the same$),$ then $28\mathrm{.}2g$

of the substance is produced. Finally, if the concentrations are changed to $6\mathrm{.}2,5\mathrm{.}7,$ and $2\mathrm{.}3\frac{g}{c}m3,$ respectively, then $29\mathrm{.}1g$ of

the chemical is produced. Find the volumes $($in cubic centimeters$)$ of the solutions containing $A,B,$ and $C.$

Solve this problem by writing a user$-$defined function using Gauss method with the matrix A and vector $B$ as input arguments

and the solution $x$ as output, $x=$Gauss$(A,B).$

Hint: write down three equations and construct the matrix A and vector $B,$ combine the Gauss forward elimination code and

back substitution code to a function, call the function to get the solution.

Note: No need of pivoting for this code

Write down $3$ equations here

$3x_1+4\mathrm{.}2x_2+6\mathrm{.}1x_3=22\mathrm{.}4$

$5\mathrm{.}4x_1+4\mathrm{.}8x_2+5\mathrm{.}5x_3=28\mathrm{.}2$

$6\mathrm{.}2x_1+5\mathrm{.}7x_2+2\mathrm{.}3x_3=29\mathrm{.}1$

$[]$ import numpy

def Gauss $($A$,$ B$)$:

#construct the matrix

#Forward Elimination

#Back Substitution

return