The ABC Ink Company is a small family owned company that sells packages of ink cartridge refills for smartpens. The Xavier set contains one blue ink refill and one black ink refill. The Yvonne set includes two blue ink refills, three black ink refills, and one red ink refill. The Zena set includes four blue ink refills, five black ink refills, and one red ink refill. The company has sold most of its stock and has found that it has only 11 blue ink cartridge refills, 14 black ink cartridge refills, and 3 red ink cartridge refills. How many of each set should the company package to sell in order to use all of the remaining ink cartridges so that there will be none left in inventory.

a. Identify the variables of the problem using x, y and z.

b. Determine the system of equations to be solved.

Solve the system of equations using Gauss-Jordan Elimination.

[tex]\left[\begin{array}{ccc}1&2&4\\1&3&5\\0&1&1\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right]=\left[\begin{array}{ccc}11\\14\\3\end{array}\right] \\\\R_2\rightarrow R_2\rightarrow R_1\\\\\left[\begin{array}{ccc}1&2&4\\0&1&1\\0&1&1\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right]=\left[\begin{array}{ccc}11\\3\\3\end{array}\right]\\\\ R_3\rightarrow R_3\rightarrow R_2\\\\[/tex]

[tex]\left[\begin{array}{ccc}1&2&4\\0&1&1\\0&0&0\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right]=\left[\begin{array}{ccc}11\\3\\0\end{array}\right]\\\\x+2y+4z=11\\y+z=3\\0=0\\\\y=3-2\\x=11-2(3-2)-4z\\=11-6+2z-4z=5-4z[/tex]

value of x and y depend upon the value of z

suppose

[tex]z=0,\,y=3,\,x=5-2z=5\\z=1,\,y=2,\,x=5-2=3\\z=2,\,y=1,\,x=5-4=1\\z=3,\,y=0,\,x=5-(2\times 3) =-1(not\,possible)[/tex]

Hence following ways company package to sell

5 xavier, 3 yvonnes , o zena or

3 xavier, 2 yvonnes, 1 zena or

1 xavier, 1 yvonnes, 2 zena