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Directions: Write a system of equations to solve the following word problem graphically and algebraically (using Gauss and using substitution).

AMC Homes, Inc. is planning to build three- and four-bedroom homes in a housing development called Chestnut Hills. Consumer demand indicates a need for three times as many four-bedroom homes as for three-bedroom homes. The net profit for each three bedroom home is $16,000 and from each four-bedroom home, $17,000. If AMC Homes must net a total profit of $13.2 million from this development, how many homes of each type should they build?

Let T and F be the respective numbers of three and four bedroom houses built.
Then, 16,000T + 17,000F = 19,200,000

We also know that the four bedroom houses are to be three times the number of 3 bedroom houses.
Therefore, F = 3T.

Substitute F = 3T into the first expression and solve for T and then F.

let T be the number of 3 bedroom homes
let F be the number of 4 bedroom homes

16000T+17000F=13200000
16000T+17000(3T)=13200000
16000T+51000T=13200000
67000T=13200000
67000T/67000=13200000/67000
T=197

Therefore, there were 197 3 bedroom homes and 591 4 bedroom homes that were built.

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