Solving Systems of Equations and Breakeven Analysis

Solving Algebraic Equations

• Solving equations with multiple variables requires having the same number of equations as there are unknowns.
  • For example, with x, y, and z, you need three equations.

Methods for Solving Equations

• One approach involves manipulating the coefficients of variables to be the same in multiple equations.

Example 1

• Given equations:
  • 5x + 4y = 7
  • 3x - 4y = 17
• Add the two equations:
  • (5x + 3x) + (4y - 4y) = 7 + 17
  • 8x = 24
• Solve for x:
  • x = \frac{24}{8} = 3
• Substitute x = 3 into one of the original equations to solve for y:
  • 5(3) + 4y = 7
  • 15 + 4y = 7
  • 4y = -8
  • y = -2
• Solution:
  • x = 3
  • y = -2

Example 2

• Given equations:
  • 4x - 3y = -7
  • 2x + 8y = 44
• Multiply the second equation by 2 to match the x coefficient in the first equation.
  • 2 * (2x + 8y) = 2 * 44
  • 4x + 16y = 88
• Multiply the second equation by -2:
  • -2 * (2x + 8y) = -2 * 44
  • -4x - 16y = -88
• Add the modified second equation to the first equation to eliminate x:
  • (4x - 4x) + (-3y - 16y) = -7 - 88
  • -19y = -95
• Solve for y:
  • y = \frac{-95}{-19} = 5
• Substitute y = 5 into one of the original equations to solve for x.

Isolating Variables

• Isolate x in the second equation:
  • 2x = 44 - 8y
  • x = 22 - 4y
• Substitute this expression for x into the first equation:
  • 4(22 - 4y) - 3y = -7
• Solve for y:
  • 88 - 16y - 3y = -7
  • -19y = -95
  • y = 5

Complex Example with Decimals

• Given equations:
  • 1.5x + 0.8y = 1.2
  • 0.7x + 1.2y = -4.4
• Multiply both equations by 10 to remove decimals:
  • 15x + 8y = 12
  • 7x + 12y = -44
• Multiply the first equation by 3 and the second equation by 2 to make the y coefficients a multiple of 24:
  • 3 * (15x + 8y) = 3 * 12
  • 45x + 24y = 36
  • 2 * (7x + 12y) = 2 * (-44)
  • 14x + 24y = -88
• Subtract the second modified equation from the first:
  • (45x - 14x) + (24y - 24y) = 36 - (-88)
  • 31x = 124
    *Solve for x:
  • x = \frac{124}{31} = 4

Equations With Fractions

• Given equations with fractions, eliminate the denominators.
• Example:
  • \frac{9x}{4} - \frac{2y}{3} = - \frac{13}{6}
  • \frac{4x}{5} + \frac{3y}{4} = \frac{123}{10}
• Multiply the first equation by 12 (LCM of 4, 3, and 6):
  • 12 * (\frac{3x}{4}) - 12 * (\frac{2y}{3}) = 12 * (-\frac{13}{6})
  • 9x - 8y = -26
• Multiply the second equation by 20 (LCM of 5, 4, and 10):
  • 20 * (\frac{4x}{5}) + 20 * (\frac{3y}{4}) = 20 * (\frac{123}{10})
  • 16x + 15y = 246
• Solving for the unknown variables
  • Multiply the first equation by 1.875
  • This will allow to easily eliminate the y variables.

Word Problems

• Word problems require translating words into algebraic equations.

General Steps

1. Assign variables: Define variables for the unknowns.
2. Create Equations: Formulate equations based on the problem.

Example 1: Car Sales

• Barbie sold twice as many cars as Ken, and together they sold 15 cars.
• Let:
  • c = number of cars Ken sold
  • 2c = number of cars Barbie sold
• Equation:
  • c + 2c = 15
  • 3c = 15
  • c = 5
  • Ken sold 5 cars and Barbie sold 10 cars.

Example 2: Daycare Purchases

• A daycare purchases milk and orange juice weekly. Prices increase, affecting the weekly bill.
• Let:
  • m = liters of milk
  • j = cans of orange juice
• Before price increase:
  • 1.10m + 0.98j = 84.40
• After price increase:
  • 1.15m + 1.14j = 91.70
• Multiply by 100 to remove decimals:
  • 110m + 98j = 8440
  • 115m + 114j = 9170
• Double check calculations by plugging them into another equation.

Example 3: Ball Purchases

• One equation is based on the total cost.
• The other equation will be based on the number of balls.

Business Applications: Breakeven Analysis

• Understanding costs, revenue, and profit.
• Determining the breakeven point.

Key Questions for Starting a Business

• Initial setup and running costs.
• Potential revenue and profit.
• Pricing strategy.
• Units to sell to break even.

Cost Types

1. Fixed Costs: Constant costs regardless of sales volume (e.g., rent, insurance).
2. Variable Costs: Costs that vary with sales volume (e.g., materials, hourly wages).

Examples of Costs

• Fixed costs: rent, salaries, property taxes.
• Variable Costs: materials, hourly wages, commissions.

Total Cost Function

• TC = FC + TVC
  • Where:
    • TC = Total Cost
    • FC = Fixed Costs
    • TVC = Total Variable Costs
• TC = FC + VC * x
  • Where:
    • VC = Variable Cost per unit
    • x = Number of units

Breakeven Point

• Total Revenue = Total Cost.
• TR - TC = 0

Profit Calculation

• Profit = Total Revenue - Total Cost.
• Profit = SP * X - (FC + VC * X)
  • Where:
    • SP = selling price
    • FC = fixed cost
    • VC = variable cost

Breakeven units

• X = \frac{FC}{SP-VC}

Breakeven Point as percentage

• \frac{Breakeven \, Amount}{Maximum \, Capacity}
  • the lower this amount is, the more potentially profitable the business is.