applications of linear equations

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McGraw-Hill Ryerson©

Applications of Linear EquationsApplications of Linear Equations

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After completing this chapter, you will be able to:

Learning Objectives

Solve two linear equations with two variables

Solve problems that require setting up linear equations with two variables

LO 2.LO 2.

LO 1.LO 1.

AlsoAlso

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Perform linear Cost-Volume-Profit and break-even analysis employing:

Learning Objectives

- The contribution margin approach

- The algebraic approach of solving the cost and revenue functions

A.A.

B.B.

LO 3.LO 3.

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2x – 6 = – 6

Solving Two Equations with Two Unknowns


LO 1.LO 1.2x – 3y = – 6 x + y = 2

EquationsEquations

(A) Solve for y2x – 3y = – 6 x + y = 4 Multiply by 2 2x + 2y = 4

Subtract - 5y = -10

y = 2y = 2 Divide by -5

(B) Solve for x(A) Solve for y

(B) Solve for x 2x – 3y = – 6 Substitute y = 22x – 3(2) = – 6

2x = + 6 – 6

x = 0 x = 0 Check…Check…

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2x – 3y = – 6 x + y = 2EquationsEquations

You should always check your answer by substituting the values into each of the equations!

x = 0 y = 2 x = 0 y = 2

= x + y= 2x – 3y



LS = RS

Left Side Right Side

LS = RS

Left Side

Equation 1Equation 1 Equation 2Equation 2

Left Side Right Side =

= – 6Substituting

= 2

Right Side

= – 6= 2(0) – 3(2) = 0 + 2

= 2

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LO 2.LO 2.

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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?

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Let x = # litres of milk Let y = # cans of orange juiceLet x = # litres of milk Let y = # cans of orange juicePurchasesPurchases

EquationsEquationsAfter price increases from $1.10 to $1.15 per litre of milk,

1.10x + 0.98y = 84.40

A.A.

B.B.

C.C.

and from$0.98 to $1.14 per can of frozen orange juice,

the weekly bill rose from $84.40 to

$91.70.

1.15x + 1.14y = 91.70

(1)

Development of…

(2)

Solving…Solving…

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Let x = # litres of milk Let y = # cans of orange juiceLet x = # litres of milk Let y = # cans of orange juice

1.10x + 0.98y = 84.40Eliminate x by Dividing

by 1.10

Eliminate x by Dividing

by 1.10

EquationEquation (1)

(1.10x + 0.98y)/1.10 = 84.40/1.10 x + 0.8909y = 76.73


1.15x + 1.14y = 91.70Eliminate x by Dividing

by 1.15

Eliminate x by Dividing

by 1.15 (1.15x + 1.14y)/1.15 = 91.70/1.15 x + 0.9913y = 79.74

…continue…continue

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EquationEquation (2) x + 0.8909y = 76.73 x + 0.9913y = 79.74

Subtract .1004y = 3.01y = 29.98 i.e. 30 cans

Substitute into

1.10x + 0.98y = 84.40EquationEquation (1)1.10x + 0.98(29.98) = 84.40

1.10x + 29.38 = 84.401.10x = 84.40 - 29.38 1.10x = 55.02

x = 50.02 i.e. 50 litres

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Cans of Orange Juice

Litres of Milk

Quantity

50

30

Price $

$1.15 $57.50

1.14 34.20

$91.70= New Weekly Cost to Purchase

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AnalysisAnalysisCostCost

LO 3.LO 3.

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TerminologyFixed CostsFixed Costs

Business Costs

Business Expenses

Variable CostsVariable Costs

…do NOT change if sales increase or

decreasee.g. rent, property taxes, some

forms of depreciation

…do change in direct proportion to sales volume e.g. material costs,

direct labour costs

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Terminology

Break Even Point

… is the point at which

neither

a Profit or Loss is made

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TerminologyContribution 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’.

…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

…is the dollar amount expressed as a percent (%) of Net Sales

A Contribution Margin statement

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$ %Net Sales(Price * # Units Sold) x 100

Less: Variable Costs x x

Net Income x x Less: Fixed Costs x x

Contribution Margin x x

TerminologyA Contribution Margin Statement

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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

Scenario 1

uestion: How much does the sale of an additional unit of a firm’s product contribute towards increasing its net income?

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Formulae Formulae

CM = S - VC

CR = CM/S * 100%

- To Find -

Contribution Margin

Contribution Rate

*Break Even Point: ...in Units (x) x = FC / CM...in Sales $ $x = (FC / CM)* S

* At Break Even, Net Profit or Loss = 0

Applying Formulae Formulae

...in % of Capacity BEPin Units/PC*100

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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.

Applying the Formulae Formulae

CM = S - VCCR = CM/S * 100%

Units x = FC / CMBreak Even Point:

In $ x = (FC / CM)* S

= $50 - $30 = $20 = $20/$50 * 100 = 40%

= $8640/$20 = 432 Units

= ($8640/$20)* $50 = $21,600= 432/ 900*100 = 48% of

CapacityBEPin units PC*100

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The Lighting Division of Seneca Electric Co. plans to introduce a new street light based on

the following accounting information:

Scenario 2

FC = $3136 VC = $157. S= $185 Capacity = 320 units

uestion: Calculate the breakeven point (BEP) …in units …in dollars …as a percent of capacity

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Scenario 2 FC = $3136 VC = $157. S= $185 Capacity = 320 units

Break Even Point…in units

…as a percent of capacity

…in dollars

= FC / CM

= (FC / CM)* S

= BEPin units/PC*100

= $3136/

S – VC = CM$185 – 157 = $28

S – VC = CM$185 – 157 = $28

28 = 112 Units

= ($3136/ 28) * $185 = $20720

= 112/320 * 100 = 35% of Capacity

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FC = $3136 VC = $157. S= $185 Capacity = 320 units

Scenario 2 -1$2688

Determine the BEP as a % of capacity if FC are reduced to $2688.

=BEPin units/PC*100 FormulaFormula

Step 1… Find CM Step 2… Find BEP in units

S = $185VC = - 157CM $ 28

= FC/CM

=

= $2688/ $28= 96 Units

=BEPin units /PC*100

Step 3… Find% of Capacity

= 96/320*100= 30% of Capacity

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FC = $3136 VC = $157 S= $185 Capacity = 320 units

Scenario 2 -2$4588

Determine the BEP as a % of capacity if FC are increased to $4588, and VC reduced to 80% of S.

= BEPin units /PC*100 FormulaFormula


S = $185VC = - 148CM $ 37

= FC/CM= $4588/ $37= 124 Units



= 124/320*100= 39% of Capacity

VC =S*80% = $148

$148

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Scenario 2 -3

Determine the BEP as a % of capacity if S is reduced to $171.

= BEPin units /PC*100 FormulaFormula


S = $ 171VC = -157CM $ 14

= FC/CM = $3136/ $14= 224 Units



= 224/320*100= 70% of Capacity

$171

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Scenario 2 -4

Determine the NI if 134 units are sold!


S = $185VC = - 157CM $ 28

= FC/CM= $3136/$28= 112 Units

UnitsSold 134BEP 112 Over BEP 22

CM of $28 per unit Company had a NI of 22* $28 = $616.

NI = #Units above BEP*CM FormulaFormula

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Scenario 2 -5

What unit sales will generate NI of $2000?

#Units above BEP = NI/CM FormulaFormula


S = $185VC = - 157CM $ 28

= FC/CM= $3136/$28= 112 Units

NI/CMNI/CM

CM of $28 per unit

= $2000/$28 per Unit

= 72 Units above Break Even

72 Units + 112 BEP Units = Total Sales Units = 184

72 Units + 112 BEP Units = Total Sales Units = 184

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Scenario 2 -6

# Units below BEP = (NI)/CM FormulaFormula


S = $185VC = - 157CM $ 28

= FC/CM= $3136/$28= 112 Units

(NI)/CM(NI)/CM

CM of $28 per unit

= 12 Units below Break Even

What are the unit sales if there is a Net Loss of $336?

= ($336)/$28 per Unit

112 BEP - 12 Units Below = Total Sales Units = 100

112 BEP - 12 Units Below = Total Sales Units = 100

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Scenario 2 -7


S = $185VC = - 157CM $ 28

= FC/CM= $3136/$28= 112 Units

CM of $28 per unit

The company operates at 85% capacity. Find the Profit or Loss.


320*.85= 272

320*.85= 272

UnitsProduction 272BEP 112 Over BEP 160

UnitsProduction 272BEP 112 Over BEP 160

# units above BEP *CM = NIFormulaFormula

160 Units * $28 = Profit $4480160 Units * $28 = Profit $4480

272

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CaseThe 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?

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CaseThe 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?What information is needed to calculate the $BEP?

2. VC per

Unit

2. VC per

Unit

1. Number of

Units sold

1. Number of

Units sold

3. CM3. CM 4. Total Costs

4. Total Costs

5. BEP in $

5. BEP in $

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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?

Case

2. VC per

Unit

2. VC per

Unit

1. Number of

Units sold

1. Number of

Units sold

3. CM3. CM

Let S = $1 and X be the Number of $1 Units soldSales of $375 000 = 375000 Total Units sold

$150000 375000

Total VCTotal Unit Sales

= = $0.40pu

S $1.00VC .40 CM $ .60

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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?

Case

4. Total Costs

4. Total Costs

5. BEP in $

5. BEP in $

TC = FC + VC = $180 000 + 0.40X

$BEP = (FC/CM)*S= ($180000/0.60)*$1.00= (300000)*$1.00# Of Units# Of Units

= $300000 $BEP

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This completes Chapter 5This completes Chapter 5

applications of linear equations

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