Two sociologists have grant money to study school busing in a particular city. They wish to conduct an opinion survey using 687 telephone contacts and 484 house contacts. Survey company A has personnel to do 31 telephone and 11 house contacts per hour; survey company B can handle 22 telephone and 16 house contacts per hour. How many hours should be scheduled for each firm to produce exactly the number of contacts needed?

We with to handle 687 telephone contacts. Company A can do 31 telephone contacts per hour, and company B can handle 22 telephone contacts per hour.

So

[tex]31x + 22y = 687[/tex]

We with to handle 484 house contacts. Company A can do 11 house contacts per hour, and company B can handle 16 house contacts per hour.

So

[tex]11x + 16y = 484[/tex]

So, we have to solve the following system of equations:

[tex]1) 31x + 22y = 687[/tex]

[tex]2) 11x + 16y = 484[/tex]

I am going to write y as a function of x in 1), and replace in 2)

[tex]31x + 22y = 687[/tex]

[tex]22y = 687 - 31x[/tex]

[tex]y = \frac{687 - 31x}{22}[/tex]

Replacing

[tex]11x + 16y = 484[/tex]

[tex]11x + 16\frac{687 - 31x}{22} = 484[/tex]

Multiplying everything by 22, we have

[tex]242x + 16\frac{687 - 31x} = 10648[/tex]

[tex]242x + 10992 - 496x = 10648[/tex]

[tex]254x = 344[/tex]

[tex]x = 1.35[/tex]

[tex]y = \frac{687 - 31x}{22} = \frac{687 - 31(1.35)}{22} = 29.3[/tex]

Company A should have 1.35 hours scheduled and company B 29.3 hours scheduled.