A cookie company makes three kinds of cookies (oatmeal raisin, chocolate chip, and shortbread) packaged in small, medium, and large boxes. The small box contains 1 dozen oatmeal raisinand 1 dozen chocolate chip; the medium box has 2 dozenoatmeal raisin, 1 dozen chocolatechip, and 1 dozenshortbread; the large boxcontains 2 dozen oatmeal raisin, 2 dozen chocolate chip, and 3 dozen shortbread. If you require exactly 15 dozen oatmeal raisin, 10 dozen chocolate chip, and 11 dozen shortbread, ow many of each size box should you buy

[tex]\begin{array}{cccc}{} & {Oatmeal} & {Chocolate} & {Shortbread} & {x} & {1} & {1} & {0} & {y} & {2} & {1} & {1}& {z} & {2} & {2} & {3} & {Total} & {15} & {10} & {11} \ \end{array}[/tex]

From the above table, we have the following equations:

Oatmeal:

[tex]x + 2y + 2z = 15[/tex] --- (1)

Chocolate Chip

[tex]x + y + 2z = 10[/tex] --- (2)

Shortbread

[tex]y + 3z = 11[/tex] --- (3)

Make y the subject in (3)

[tex]y =11 - 3z[/tex]

Substitute [tex]y =11 - 3z[/tex] in (1) and (2)

[tex]x + 2y + 2z = 15[/tex] --- (1)

[tex]x + 2(11-3z) + 2z = 15[/tex]

[tex]x + 22 - 6z + 2z = 15[/tex]

[tex]x -6z +2z = 15 - 22\\[/tex]

[tex]x-4z = -7[/tex]

[tex]x = 4z - 7[/tex] --- (4)

[tex]x + y + 2z = 10[/tex] --- (2)

[tex]x + 11 - 3z + 2z = 10[/tex]

[tex]x - 3z + 2z = 10 - 11[/tex]

[tex]x -z = -1[/tex]

[tex]x = z - 1[/tex] --- (5)

Equate (4) and (5)

[tex]4z - 7 = z - 1[/tex]

[tex]4z - z = 7 - 1[/tex]

[tex]3z = 6[/tex]

[tex]z = 2[/tex]

Substitute [tex]z = 2[/tex] in (5)

[tex]x = z - 1[/tex] --- (5)

[tex]x =2-1[/tex]

[tex]x =1[/tex]

Substitute [tex]z = 2[/tex] in [tex]y =11 - 3z[/tex]

[tex]y = 11 - 3 * 2[/tex]

[tex]y = 11 - 6[/tex]

[tex]y = 5[/tex]

So, the solution is:

[tex]x =1[/tex] [tex]y = 5[/tex] [tex]z = 2[/tex]