W3 Busn Math - 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.