. 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.

The company must use of of the Xavier's product in order to use the leftover stock to package 3 Zena's Product.

Step-by-step explanation:

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.

Let the products be:

Xavier(x)

Yvonne(y)

Zena(z).

For the Blue Cartridge

Xavier Uses 1, Yvonne uses 2 and Zena uses 4.

1x+2y+4z

For the Black Cartridge

Xavier Uses 1, Yvonne uses 3 and Zena uses 5.

1x+3y+5z

For the Red Cartridge

Xavier Uses 0, Yvonne uses 4 and Zena uses 1.

0x+4y+1z

Inventory Left

Blue= 11 ink cartridge refills,

Black= 14 ink cartridge refills,

Red= 3 ink cartridge refills.

Therefore, in order to use all the cartridges left:

1x+2y+4z=11

x+3y+5z=14

4y+z=3

From equation 3

z=3-4y

Substitute into equations 1 and 2

1x+2y+4z=11

x+2y+4(3-4y)=11

x+2y+12-16y=11

x-14y=-1

x+3y+5z=14

x+3y+5(3-4y)=14

x+3y+15-20y=14

x-17y=-1

Solving the resulting equations

x-14y=-1

x-17y=-1

3y=0

y=0

Substituting y=0 into x-17y=-1

x=-1

Recall:

z=3-4y

z=3.

x=-1, y=0, z=3.

The company must use of of the Xavier's product in order to use the leftover stock to package 3 Zena's Product.

Note: You can use Gaussian Elimination Method to solve the system of linear equations.