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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 $110 per square foot for the cabin.

The second contractor gives a quote of $110,000 plus $70 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 = 30,000 + 110f[/tex] represents the cost charged by the first contractor.

The equation [tex]C = 110,000 + 70f[/tex] represents the cost charged by the second contractor.

Answer :

The equation C= 30000 + 110f = 110000 + 70f represents the cost charged by the first contractor. This solves to 250,000.

To solve this problem, we will set up two linear equations representing the total cost for each contractor. Let f represent the square feet and C represent the cost.

Step-by-Step Explanation:

  • The first contractor's quote can be expressed as: C = 30000 + 110f
  • The second contractor's quote can be expressed as: C = 110000 + 70f

We need to find the value of f for which the two cost expressions are equal, meaning we need to solve for when 30000 + 110f = 110000 + 70f.

By isolating the variable f, we can solve the equation:

  • Subtract 70f from both sides: 30000 + 110f - 70f = 110000

  • This simplifies to: 30000 + 40f = 110000

  • Subtract 30000 from both sides: 40f = 80000

  • Divide both sides by 40: f = 2000

So, the two contractors will charge the same amount for a cabin of 2000 square feet.

Next, we substitute f = 2000 back into either cost equation to find the cost:

First contractor's cost: C = 30000 + 110(2000) = 30000 + 220000 = 250000

The cost for both contractors will be 250,000.

Therefore, for a cabin of 2000 square feet, both contractors will charge 250,000.

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