Understanding Linear Equations

Concepts of Linear Equations

• An algebraic expression combines numbers and variables through addition, subtraction, multiplication, division, exponentiation, or extracting roots
• An algebraic sum has parts connected by plus or minus signs
• Terms are each part of an algebraic sum, along with its preceding sign
• Coefficients are numerical coefficients of terms
• An equation shows that two algebraic expressions are equal
• A linear equation with two variables is in the form ax + by = c, where a, b, and c are fixed real numbers, and x and y are variables

Linear Equations - Restrictions

• In the equation of a line, a and b can't both be zero simultaneously
• If a = 0 and b = 0, then 0 = c, which is either true or false
• Variables' powers are one, and variables are multiplied by a constant
• The degree (power) of the variables is 1 in a linear equation
• Linear equations in two variables have a constant slope throughout the line

Slope Intercept Summary

• y = mx + b represents a linear equation where:
  • y is the dependent variable
  • x is the independent variable
  • m is the slope
  • b is the y-intercept

Slope Calculation

• Slope (m) is the change in y over the change in x, also known as rise over run
• m = (y2 - y1) / (x2 - x1) if (x1, y1) and (x2, y2) are two distinct points on the line
• Slope measures the steepness of a line
  • A larger positive slope indicates a steeper line
  • A smaller negative slope indicates a shallower line

Lines and Slope

• A line parallel to the x-axis has a slope of 0
• A line parallel to the y-axis has an undefined slope

Slope Definition

• The slope of a line is the change along the vertical axis relative to the change along the horizontal axis
• In business, slope is the change in the dependent variable y relative to a one-unit change in the independent variable x

Developing a Linear Equation

• A line's equation can be found with enough information
• Two points uniquely determine the equation of a line that passes through them

Ways to Develop a Linear Equation

• Two-point form
• Slope-point form
• Slope-intercept form

Two-Point Form

• Two distinct points on a line determine a unique straight line
• Given two points P1(x1,y1) and P2(x2,y2)
• The equation is (y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)

Slope-Point Form

• Uses the slope (m) and a point P1(x1, y1) on the line
• The equation is y - y1 = m(x - x1)

Slope-Intercept Form

• Uses the slope (m) and the y-intercept b of the line
• Since the y-intercept is the point where x = 0, it has coordinates P1(0, b)
• The equation of the line is y = mx + b

Application of Linear Equations

• Used in Business, Economics, Management, as well as in the social and natural sciences
• Simplifies understanding relationships between variables algebraically and geometrically
• Useful in cost-output relationship, break-even analysis, and merchandising

Linear Cost Output Relationships

• Total cost (TC) of production is linearly related to the number of units of products (Q) produced and sold
• VC = variable cost per unit
• FC = fixed costs
• TVC = total variable costs
• AC = average cost
• MC = marginal cost
• TR = total revenue
• Π = profit

Cost Equation Summary

• TC = TFC + TVC
• TR = unit price × quantity = P.Q
• π = TR - TC
• π = P.Q - (VC + FC)
• π = PQ - Q.VC - FC
• π = Q. (P – VC) – FC

Graph Interpretations

• Vertical distance between AB, FC, GD is the same because Fixed Cost is constant
• If there are no sales, there is no revenue, hence the total revenue function passes through the origin
• Without production, fixed costs exist
• Total cost function starts from the fixed cost point on the graph rather than the origin
• Up to point T, total costs exceed revenues, resulting in a loss
• At point T, TR = TC, indicating the break-even point where profit is zero
• To the right of point T, TR > TC meaning there is profit or gain
• Fixed costs remain constant at all levels of production
• As production increases, total variable cost (TVC) and unit variable cost (MC) increase at the same rate
• As production increases, TC increases at the rate equal to average variable cost (AVC) where AVC = MC

Cost Relationships

• AVC remains constant across all production levels
• Average Fixed Cost (AFC) decreases when Quantity increases
• Average Total Cost (ATC) decreases as Quantity increases due to the effect of the decrease in AFC
• As Quantity increases, TR increases at a rate of unit sales price P
• Average revenue remains constant
• AR = P, provided the revenue function is linear

Breakeven Analysis

• Observing the level of production that leads from loss to profit is important
• Break-even point is where there is neither profit nor loss
• It can be expressed as production quantity or revenue level

Cost Equations

• Manufacturing companies state cost in terms of quantity produced and sold
• Retail businesses express cost in terms of revenue

Manufacturing Cost Companies

• Total cost = Fixed cost + total variable cost
• TC = FC + TVC
• Total Revenue = unit price × quantity
• TR = P.Q
• At Break-even point, TR = TC, i.e. TR – TC = 0
• P.Q = VC + FC
• P.Q – VC.Q = FC
• Q.(P – VC) = FC
• Qe = FC / (P – VC), where Qe is Breakeven Quantity
• The revenue at the break-even quantity is the break-even revenue (BER)

Altering One Variable

• Fixed costs (FC) and break-even quantity (Qe) have a direct relationship
• Unit Variable cost (VC) and break-even quantity (Qe) have a direct relationship
• Unit price (P) and break-even points (Qe) have an indirect or inverse relationship

Company Strategies

• To minimize the breakeven point and maximize profit companies should:
  • Decrease its fixed costs
  • Decrease its unit variable cost
  • Increase the unit selling price
• It's preferable to decrease the unit variable cost, of those options since increased selling price can result in losing customers

Finding Quantity Levels

• Π = TR – TC
• П = PQ – (VC.Q + FC)
• П = (P.Q – VC.Q) – FC
• П = Q (P – VC) – FC
• Q = (FC ± π) / (P – VC), for any level of production and sales Q
• If there is a gain, then the profit  is positive, and if there is a loss then the profit  is negative

Merchandising

• Break-even analysis is considered with respect to manufacturing companies that state their cost equation in terms of quantity
• Break-even analysis with respect to retail businesses state their cost equation in terms of revenue

Revenue Definitions

• Break-even Revenue = BEQ X P
• Markup = Selling price – Variable cost
• The markup as a function of cost is: cost / 50%
• As a function of retail price, the markup is also margin (33.3%)
• The cost of goods sold is approximately 67%

Retail Sales Equations

• Given selling expense = 1% of sales
• The TC equation becomes Y = 0.68X + FC
  • Y is the total cost
  • X is sales revenue
• 68% is the variable cost of goods purchased and sold
• Total cost for the retailer is given by Y = mX + b
  • m is the variable cost per unit

Breakeven Revenue and General Information

• Break-even revenue (BER) is obtained by making sales revenue and cost equal
• FC Xe = or Xe = (FC + Π) / (1 – m), where m = VC / P = TVC / TR 1-m
• Break-even revenue (BER) method is useful because a single formula can be used for different goods if the company uses the same profit margin for all goods
• In Break-even quantity method or BEQ, this is not possible and requires a different formula.