Solving Problems Using Systems of Linear Equations

Introduction to Linear Equations in Business

• Concept: Solving problems using systems of linear equations.
• Application: Determining profit, revenue, and cost.

Key Definitions

• Revenue (R): Money generated by selling x units of a product.
• Cost (C): Money spent producing x units of a product.
• Profit: Revenue minus cost.

Revenue and Cost Functions

• Revenue Function (R of x):
  • Formula: R(x) = price per unit sold * x
• Cost Function (C of x):
  • Formula: C(x) = fixed cost + (cost per unit produced * x)

Break-Even Point

• Definition: The point where revenue equals cost.
• Graphically: Intersection of revenue and cost functions.
• Coordinates:
  • x-coordinate: Number of units where revenue equals cost.
  • y-coordinate: Amount of money coming in and going out.

Example: Manufacturing Wheelchairs

Problem Setup

• Fixed Costs: $500,000
• Cost per Unit: $400
• Selling Price per Unit: $600

Cost Function

• Formula: C, of producing (x) wheelchairs = Fixed cost 500,000 + Variable Cost 400x
• C(x) = 500,000 + 400x

Revenue Function R, of producing x wheelchairs. Revenue per chair, $600 x number of chairs sold.

• Formula: R(x) = 600x

Finding the Break-Even Point. Since we want point where revenue and cost equal, we’ll let both equal to y, giving us y = 500,000 + 400x, and y = 600x.

• System of Equations:
  • C(x) = 500,000 + 400x
  • R(x) = 600x
• Substitution Method: Substitute 600x for y in first equation to get 600x = 500,000 + 400x.
  • Set R(x) = C(x)
  • Solve: 600x = 500,000 + 400x
  • Simplify: 200x = 500,000(subtracting 400x from both sides)
  • Solution: x = 2,500
  • Since revenue and cost equal, can use either equation. R(x) = 600x and substitute 2,500 for x giving R (2500) = 600 (2500) = 1,500,000.

Interpretation

• Break-Even Point: ordered pair (2,500, 1,500,000)
• Meaning: The company breaks even by producing and selling 2,500 wheelchairs, with revenue and cost both at $1,500,000. (All this level money coming in = money going out.)

Profit Function Profit generated by business is money taken in (revenue) - money spent (cost). Profit P(x), generated after producing and selling x units of product is given by profit function.

• Formula: P(x) = R(x) - C(x)- where R and C are revenue and cost functions, respectively.
• Purpose: Determines profit function (P(x) after modeling revenue and cost with system of equations.

Conclusion

• Understanding these concepts helps businesses determine when they will start making a profit, which is crucial for strategic planning and decision-making.