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Applications of Linear Equations

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

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Applications of Linear Equations

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

  2. Learning Objectives After completing this chapter, you will be able to: Solvetwolinear equationswith two variables LO 1. LO 2. Solveproblems that require setting up linear equations withtwo variables Also

  3. Learning Objectives Perform linearCost-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.

  4. Substitute y = 2 Multiply by 2 Subtract Divide by -5 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 2x + 2y = 4 -5y = -10 y = 2 (B) Solve for x 2x – 3(2) = – 6 2x – 3y = – 6 2x – 6 = – 6 2x = + 6 – 6 Check… x = 0

  5. 2x – 3y = – 6 x + y = 2 Equations = LeftSide RightSide Show Note Substituting = 2 = – 6 LS =RS LS =RS Solving Two Equations with Two Unknowns You should always checkyour answer by substituting the valuesinto each of the equations! x = 0y=2 Equation 1 Equation 2 LeftSide RightSide LeftSide RightSide = 2x – 3y = x + y = 2(0) – 3(2) = 0 + 2 = – 6 = 2

  6. Setting up linear equations with Two Variables LO 2.

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

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

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

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

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

  12. LO 3. Cost Analysis

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

  14. Terminology BreakEvenPoint … is the point at which neither a Profit or Loss is made

  15. Terminology A Contribution Margin statement 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

  16. Terminology A Contribution Margin Statement $ % NetSales(Price * # Units Sold) x100 Less: Variable Costs x x Contribution Margin x x Less: Fixed Costs x x Net Income x x

  17. Scenario 1 Q uestion: 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 How much does the sale of an additional unit of a firm’s product contributetowards increasing its net income?

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

  19. Applying the Formulae 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 $30per unit, and theProductionCapacity is 900 units per period. CM = S - VC = $50 - $30 = $20 CR=CM/S* 100% = $20/$50 * 100 = 40% BreakEven Point: Unitsx =FC / CM = $8640/$20 = 432 Units In $ x =(FC / CM)*S = ($8640/$20)* $50 = $21,600 BEPin unitsPC*100 = 432/ 900*100 = 48% of Capacity

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

  21. Scenario 2 S – VC = CM $185 – 157 = $28 FC= $3136VC= $157.S= $185 Capacity = 320 units Break Even Point …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

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

  23. Scenario 2 -2 $4588 Formula = BEPin units /PC*100 FC= $3136VC= $157S= $185Capacity = 320units $148 VC =S*80% = $148 Determine the BEP as a % of capacity if FC are increased to $4588, and VC reduced to80%ofS. Step 3… Find % of Capacity Step 2… FindBEP in units Step 1… FindCM =BEPin units /PC*100 S = $185 = FC/CM VC= - 148 CM $ 37 = $4588/$37 = 124/320*100 = 124 Units = 39% of Capacity

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

  25. Scenario 2 -4 Formula NI = #Units aboveBEP*CM FC= $3136VC= $157S= $185Capacity = 320units Determine the NI if 134 units are sold! Units Sold 134 BEP 112 OverBEP 22 Step 2… FindBEP in units Step 1… FindCM 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.

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

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

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

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

  30. Case The Marconi Co. year end operating results were as follows: Total Sales of $375000 Operated at 75% of capacity TotalVariable Costs were $150000 TotalFixed 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. VCper Unit 3. CM 4. Total Costs 5. BEP in $

  31. Case The Marconi Co. year end operating results were as follows: Total Sales of $375000 Operated at 75% of capacity TotalVariable Costs were $150000 TotalFixed 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 $150000 375000 2. VCper Unit Total VC Total Unit Sales = $0.40pu = S $1.00 VC .40 CM $ .60 3. CM

  32. Case The Marconi Co. year end operating results were as follows: Total Sales of $375000 Operated at 75% of capacity TotalVariable Costs were $150000 TotalFixed Costs were $180000 What was Marchoni’s BEP expressed in dollars of sales? # Of Units = $300000 $BEP 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

  33. This completes Chapter 5

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