Applications of Linear Equations If...X+5 = $200 What does X =? Applications of Linear Equations Chapter 5

Solve two linear equations with two variables Learning Objectives After completing this chapter, you will be able to: Solve two linear equations with two variables LO 1. LO 2. Solve problems that require setting up linear equations with two variables Also

Learning Objectives Perform linear Cost-Volume-Profit and break-even analysis employing: LO 3. - The contribution margin approach A. - The algebraic approach of solving the cost and revenue functions B.

Solving Two Equations with Two Unknowns LO 1. Equations 2x – 3y = – 6 x + y = 2 (A) Solve for y (B) Solve for x (A) Solve for y 2x – 3y = – 6 x + y = 4 Multiply by 2 2x + 2y = 4 - 5y = -10 Subtract y = 2 Divide by -5 (B) Solve for x 2x – 3(2) = – 6 2x – 3y = – 6 Substitute y = 2 2x – 6 = – 6 2x = + 6 – 6 Check… x = 0

Solving Two Equations with Two Unknowns 2x – 3y = – 6 x + y = 2 Equations Note You should always check your answer by substituting the values into each of the equations! x = 0 y = 2 Left Side Right Side = Show Equation 1 Equation 2 Left Side Right Side Left Side Right Side = 2x – 3y = – 6 = 2 = x + y = 2(0) – 3(2) Substituting = 0 + 2 = – 6 = 2 LS = RS LS = RS

Setting up linear equations with Two Variables LO 2.

York Daycare purchases the same amount of Setting up linear equations with Two Variables Q York Daycare purchases the same amount of milk and orange juice each week. After price increases from $1.10 to $1.15 per litre for milk, and from $0.98 to $1.14 per can of frozen orange juice, the weekly bill rose from $84.40 to $91.70. How many litres of milk and cans of orange juice are purchased each week?

Setting up linear equations with Two Variables A. B. C. Purchases Let x = # litres of milk Let y = # cans of orange juice Equations After price increases from $1.10 to $1.15 per litre of milk, A. Development of… and from $0.98 to $1.14 per can of frozen orange juice, (1) 1.10x + 0.98y = 84.40 B. (2) 1.15x + 1.14y = 91.70 the weekly bill rose from $84.40 to $91.70. C. Solving…

Let x = # litres of milk Let y = # cans of orange juice Setting up linear equations with Two Variables Let x = # litres of milk Let y = # cans of orange juice Equation (1) Eliminate x by Dividing by 1.10 1.10x + 0.98y = 84.40 (1.10x + 0.98y)/1.10 = 84.40/1.10 x + 0.8909y = 76.73 Equation (2) Eliminate x by Dividing by 1.15 1.15x + 1.14y = 91.70 (1.15x + 1.14y)/1.15 = 91.70/1.15 x + 0.9913y = 79.74 …continue

Setting up linear equations with Two Variables Proof (1) x + 0.8909y = 76.73 x + 0.9913y = 79.74 Equation (2) .1004y = 3.01 Subtract y = 29.98 i.e. 30 cans 1.10x + 0.98y = 84.40 Equation (1) Substitute into 1.10x + 0.98(29.98) = 84.40 1.10x + 29.38 = 84.40 1.10x = 84.40 - 29.38 1.10x = 55.02 x = 50.02 i.e. 50 litres Proof

Setting up linear equations with Two Variables Proof Litres of Milk 50 Quantity Price $ Litres of Milk 50 $1.15 $57.50 Cans of Orange Juice 30 1.14 34.20 $91.70 = New Weekly Cost to Purchase

LO 3. Cost Analysis

Terminology or Business Costs Business Expenses Fixed Costs …do NOT change if sales increase or decrease e.g. rent, property taxes, some forms of depreciation Variable Costs …do change in direct proportion to sales volume e.g. material costs, direct labour costs

… is the point at which neither a Profit or Loss is made Terminology Break Even Point … is the point at which neither a Profit or Loss is made

…is the dollar amount expressed as a percent (%) of Net Sales Terminology Contribution Margin …is the dollar amount that is found by deducting ALL Variable Costs from Net Sales and ‘contributes’ to meeting Fixed Costs and making a ‘Net Profit’. Contribution Rate …is the dollar amount expressed as a percent (%) of Net Sales A Contribution Margin statement

Terminology A Contribution Margin Statement $ % $ % Net Sales(Price * # Units Sold) x 100 Less: Variable Costs x x Contribution Margin x x Less: Fixed Costs x x Net Income x x

Scenario 1 Market research for a new product indicates that the product can be sold at $50 per unit. Cost analysis provides the following information: Fixed Costs per period = $8640 Variable Costs = $30 per unit. Production Capacity per period = 900 units Q uestion: How much does the sale of an additional unit of a firm’s product contribute towards increasing its net income?

Formulae Formulae - To Find - x = FC / CM $x = (FC / CM)* S Contribution Margin CM = S - VC Contribution Rate CR = CM/S * 100% *Break Even Point: ...in Units (x) x = FC / CM $x = (FC / CM)* S ...in Sales $ ...in % of Capacity BEPin Units/PC*100 * At Break Even, Net Profit or Loss = 0 Applying Formulae

Applying the Formulae CM = S - VC = $50 - $30 = $20 CR = CM/S * 100% As in the previous scenario, the new product can be sold at $50 per unit. Costs are as follows: Fixed Costs are $8640 for the period , Variable Costs are $30 per unit, and the Production Capacity is 900 units per period. CM = S - VC = $50 - $30 = $20 CR = CM/S * 100% = $20/$50 * 100 = 40% Break Even Point: Units x = FC / CM = $8640/$20 = 432 Units In $ x = (FC / CM)* S = ($8640/$20)* $50 = $21,600 BEPin units PC*100 = 432/ 900*100 = 48% of Capacity

Scenario 2 Q Calculate the breakeven point (BEP) …in units …in dollars The Lighting Division of Seneca Electric Co. plans to introduce a new street light based on the following accounting information: FC = $3136 VC = $157. S= $185 Capacity = 320 units Q uestion: Calculate the breakeven point (BEP) …in units …in dollars …as a percent of capacity

Scenario 2 Break Even Point = FC / CM …in units …in dollars FC = $3136 VC = $157. S= $185 Capacity = 320 units Break Even Point S – VC = CM $185 – 157 = $28 …in units = FC / CM = $3136/ 28 = 112 Units …in dollars = (FC / CM)* S = ($3136/ 28) * $185 = $20720 …as a percent of capacity = BEPin units/PC*100 = 112/320 * 100 = 35% of Capacity

Determine the BEP as a % of capacity if FC are reduced to $2688. Scenario 2 -1 FC = $3136 VC = $157. S= $185 Capacity = 320 units $2688 Determine the BEP as a % of capacity if FC are reduced to $2688. Formula =BEPin units/PC*100 Step 2… Find BEP in units Step 3… Find % of Capacity Step 1… Find CM =BEPin units /PC*100 S = $185 = FC/CM VC = - 157 CM $ 28 = $2688/ $28 = 96/320*100 = 96 Units = 30% of Capacity =

Scenario 2 -2 Formula Step 2… Find BEP in units Step 3… Find FC = $3136 VC = $157 S= $185 Capacity = 320 units $4588 $148 VC =S*80% = $148 Determine the BEP as a % of capacity if FC are increased to $4588, and VC reduced to 80% of S. = BEPin units /PC*100 Formula Step 2… Find BEP in units Step 3… Find % of Capacity Step 1… Find CM =BEPin units /PC*100 S = $185 = FC/CM VC = - 148 CM $ 37 = $4588/ $37 = 124/320*100 = 124 Units = 39% of Capacity

Determine the BEP as a % of capacity if S is reduced to $171. Scenario 2 -3 FC = $3136 VC = $157 S= $185 Capacity = 320 units $171 Determine the BEP as a % of capacity if S is reduced to $171. = BEPin units /PC*100 Formula Step 2… Find BEP in units Step 3… Find % of Capacity Step 1… Find CM =BEPin units /PC*100 S = $ 171 = FC/CM VC = -157 CM $ 14 = $3136/ $14 = 224/320*100 = 224 Units = 70% of Capacity

Determine the NI if 134 units are sold! Scenario 2 -4 FC = $3136 VC = $157 S= $185 Capacity = 320 units Determine the NI if 134 units are sold! NI = #Units above BEP*CM Formula Step 2… Find BEP in units Units Sold 134 BEP 112 Over BEP 22 Step 1… Find CM S = $185 = FC/CM VC = - 157 CM $ 28 = $3136/$28 = 112 Units CM of $28 per unit Company had a NI of 22 * $28 = $616.

What unit sales will generate NI of $2000? = 72 Units above Break Even Scenario 2 -5 FC = $3136 VC = $157 S= $185 Capacity = 320 units What unit sales will generate NI of $2000? #Units above BEP = NI/CM Formula Step 1… Find CM Step 2… Find BEP in units S = $185 VC = - 157 CM $ 28 = FC/CM = $3136/$28 = 112 Units NI/CM = $2000/$28 per Unit = 72 Units above Break Even CM of $28 per unit 72 Units + 112 BEP Units = Total Sales Units = 184

What are the unit sales if there is a Net Loss of $336? Scenario 2 -6 FC = $3136 VC = $157 S= $185 Capacity = 320 units What are the unit sales if there is a Net Loss of $336? # Units below BEP = (NI)/CM Formula Step 1… Find CM Step 2… Find BEP in units S = $185 VC = - 157 CM $ 28 = FC/CM = $3136/$28 = 112 Units (NI)/CM = ($336)/$28 per Unit = 12 Units below Break Even CM of $28 per unit 112 BEP - 12 Units Below = Total Sales Units = 100

The company operates at 85% capacity. Find the Profit or Loss. Scenario 2 -7 FC = $3136 VC = $157 S= $185 Capacity = 320 units 272 The company operates at 85% capacity. Find the Profit or Loss. 320*.85 = 272 Step 1… Find CM Step 2… Find BEP in units S = $185 VC = - 157 CM $ 28 = FC/CM = $3136/$28 = 112 Units Units Production 272 BEP 112 Over BEP 160 CM of $28 per unit # units above BEP *CM = NI Formula 160 Units * $28 = Profit $4480

Case The Marconi Co. year end operating results were as follows: Total Sales of $375000 Operated at 75% of capacity Total Variable Costs were $150000 Total Fixed Costs were $180000 What was Marconi’s BEP expressed in dollars of sales?

Case The Marconi Co. year end operating results were as follows: Total Sales of $375000 Operated at 75% of capacity Total Variable Costs were $150000 Total Fixed Costs were $180000 What was Marconi’s BEP expressed in dollars of sales? What information is needed to calculate the $BEP? 1. Number of Units sold 2. VC per Unit 3. CM 4. Total Costs 5. BEP in $

Case = = $0.40pu 1. Number of Units sold The Marconi Co. year end operating results were as follows: Total Sales of $375000 Operated at 75% of capacity Total Variable Costs were $150000 Total Fixed Costs were $180000 What was Marchoni’s BEP expressed in dollars of sales? 1. Number of Units sold Let S = $1 and X be the Number of $1 Units sold Sales of $375 000 = 375000 Total Units sold 2. VC per Unit Total VC Total Unit Sales $150000 375000 = = $0.40pu S $1.00 VC .40 CM $ .60 3. CM

Case $BEP = (FC/CM)*S = ($180000/0.60)*$1.00 = (300000)*$1.00 The Marconi Co. year end operating results were as follows: Total Sales of $375000 Operated at 75% of capacity Total Variable Costs were $150000 Total Fixed Costs were $180000 What was Marchoni’s BEP expressed in dollars of sales? 4. Total Costs TC = FC + VC = $180 000 + 0.40X $BEP = (FC/CM)*S 5. BEP in $ = ($180000/0.60)*$1.00 = (300000)*$1.00 # Of Units = $300000 $BEP

This completes Chapter 5