The Enormous State University's Business School is buying computers. The school has two models to choose from, the Pomegranate and the iZac. Each Pomegranate comes with 400 GB of memory and 80 TB of disk space, while each iZac has 300 GB of memory and 100 TB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least 52,500 GB of memory and at least 12,700 TB of disk space. If both the Pomegranate and the iZac cost $2,200 each, how many of each should the school buy to keep the cost as low as possible?

[tex]\begin{array}{cccc}{} & {Pomegranate} & {iZac} & {Total} & {Memory} & {400} & {300} & {52500} & {Disk\ Space} & {80} & {100} & {12700} \ \end{array}[/tex]

The table can be represented as follows:

[tex]400p + 300i = 52500[/tex] --- (1)

[tex]80p + 100i = 12700[/tex] ---- (2)

Multiply equation (2) by 3

[tex]3 *\ ([/tex][tex]80p + 100i = 12700[/tex][tex])[/tex]

[tex]240p + 300i = 38100[/tex] ---- (3)

Subtract (3) from (1)

[tex]400p - 240p + 300i - 300i = 52500 - 38100[/tex]

[tex]160p = 14400[/tex]

Divide both sides by 160

[tex]p = 90[/tex]

Substitute [tex]p = 90[/tex] in (2)

[tex]80p + 100i = 12700[/tex]

[tex]80 * 90 + 100i = 12700[/tex]

[tex]7200 + 100i = 12700[/tex]

Collect like terms

[tex]100i = 12700 - 7200[/tex]

[tex]100i = 5500[/tex]

Divide by 100

[tex]i = 55[/tex]

So;

[tex]p = 90[/tex] and [tex]i = 55[/tex]