Set up a system of linear equations in two variables that models the problem. Then solve the system of linear equations.

A person plans to build a cabin at the lake and wants to get bids from two contractors to determine its cost. The first contractor gives a quote of $50,000 plus $100 per square foot for the cabin. The second contractor gives a quote of $70,000 plus $90 per square foot. For how many square feet will the two contractors charge the same amount? What is that cost?

Let [tex]f[/tex] represent the square feet and [tex]C[/tex] represent the cost charged by the contractor. (Do not include the $ symbol in your answers.)

The equation [tex]C = 100f + 50000[/tex] represents the cost charged by the first contractor.

The equation [tex]C = 90f + 70000[/tex] represents the cost charged by the second contractor.

They, at 2000 square feet, both contractors will charge 250,000. To find the cost at this square footage, we can substitute f = 2000 into either of the original equations.

The equation for the cost charged by the first contractor can be represented as:
C1 = 50000 + 100f

The equation for the cost charged by the second contractor can be represented as:
C2 = 70000 + 90f

Now, let's solve the system of equations to find the square footage where the two contractors charge the same amount. We can do this by setting the two equations equal to each other and solving for f:

50000 + 100f = 70000 + 90f

First, let's simplify the equation by combining like terms:
10f = 20000

Next, we'll isolate f by dividing both sides of the equation by 10:
f = 2000

For example, using the equation for the first contractor:
C1 = 50000 + 100(2000)
C1 = 50000 + 200000
C1 = 250000

The, at 2000 square feet, both contractors will charge 250,000.

In conclusion, the system of linear equations that models this problem is:
C1 = 50000 + 100f
C2 = 70000 + 90f

At 2000 square feet, both contractors will charge 250,000.

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