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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 $30,000 plus $100 per square foot for the cabin.
- The second contractor gives a quote of $50,000 plus $70 per square foot.

For how many square feet will the two contractors charge the same amount? What is that cost?

Let \( f \) represent the square feet and \( C \) represent the cost charged by the contractor.

The equation \( C = 30,000 + 100f \) represents the cost charged by the first contractor.

The equation \( C = 50,000 + 70f \) represents the cost charged by the second contractor.

Answer :

The cost charged by both contractors for a 1500 square foot cabin is $155,000.

To set up a system of linear equations for this problem, let's use the following variables:

- f: square feet of the cabin

- C1: cost charged by the first contractor

- C2: cost charged by the second contractor

According to the information provided, we can write the equations:

For the first contractor:

C1 = $5000 + $100f

For the second contractor:

C2 = $50000 + $70f

To obtain the number of square feet for which the two contractors charge the same amount, we need to solve the system of equations:

C1 = C2

Substituting the expressions for C1 and C2, we have:

$5000 + $100f = $50000 + $70f

To solve for f, we can simplify the equation:

$100f - $70f = $50000 - $5000

$30f = $45000

Dividing both sides by $30, we get:

f = $45000 / $30

Simplifying, we obtain:

f = 1500 square feet

Therefore, the two contractors will charge the same amount for a cabin with 1500 square feet.

To obtain the cost, we can substitute this value of f into either of the original equations:

C1 = $5000 + $100(1500)

= $5000 + $150000

= $155000

To know more about system of linear equations refer here:

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