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Applications of Linear Equations

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If...X+5 = $200

What does X =?

Applications of Linear Equations

Chapter 5

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

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.

Substitute y = 2

Multiply by 2

Subtract

Divide by -5

Solving Two Equations with Two Unknowns

LO 1.

Equations

2x â€“ 3y = â€“ 6

x + y = 2

(A) Solve for y

(B) Solve for x

(A) Solve for y

2x â€“ 3y = â€“ 6

x + y = 4

2x + 2y = 4

-5y = -10

y = 2

(B) Solve for x

2x â€“ 3(2) = â€“ 6

2x â€“ 3y = â€“ 6

2x â€“ 6 = â€“ 6

2x = + 6 â€“ 6

Checkâ€¦

x = 0

2x â€“ 3y = â€“ 6

x + y = 2

Equations

=

LeftSide

RightSide

Show

Note

Substituting

= 2

= â€“ 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 â€“ 3y

= x + y

= 2(0) â€“ 3(2)

= 0 + 2

= â€“ 6

= 2

Setting up

linear equations

with Two Variables

LO 2.

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?

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 ofâ€¦

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.

Solvingâ€¦

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

â€¦continue

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

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

LO 3.

Cost

Analysis

Terminology

Fixed Costs

Business Costs

Business Expenses

or

Variable Costs

â€¦do NOT change if sales increase or decrease

e.g. rent, property taxes, some forms of depreciation

â€¦do change in direct proportion to sales volume e.g. material costs, direct labour costs

Terminology

BreakEvenPoint

â€¦ is the point at which neither a Profit or Loss is made

Terminology

A Contribution Margin statement

Contribution Margin

â€¦is the dollar amount that is found by deducting ALL Variable Costs from Net Sales and â€˜contributesâ€™ to meeting Fixed Costs and making a â€˜Net Profitâ€™.

Contribution Rate

â€¦is the dollar amount expressed as a percent (%) of Net Sales

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

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 firmâ€™s product contributetowards increasing its net income?

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

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

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)

â€¦in units

â€¦in dollars

â€¦as a percent of capacity

Scenario 2

S â€“ VC = CM

$185 â€“ 157 = $28

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

Break Even Point

â€¦in units

=FC / CM

= $3136/

28 =

112 Units

â€¦in dollars

=(FC / CM)*S

= ($3136/

28) *

$185 = $20720

â€¦as a percent of capacity

= BEPin units/PC*100

= 112/320 * 100 = 35% of Capacity

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 3â€¦ Find

% of Capacity

Step 2â€¦ FindBEP in units

Step 1â€¦ FindCM

=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

$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 3â€¦ Find

% of Capacity

Step 2â€¦ FindBEP in units

Step 1â€¦ FindCM

=BEPin units /PC*100

S = $185

= FC/CM

VC= - 148

CM $ 37

= $4588/$37

= 124/320*100

= 124 Units

= 39% of Capacity

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 3â€¦ Find

% of Capacity

Step 2â€¦ FindBEP in units

Step 1â€¦ FindCM

=BEPin units /PC*100

S = $ 171

= FC/CM

VC= -157

CM $ 14

= $3136/$14

= 224/320*100

= 224 Units

= 70% of Capacity

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 2â€¦ FindBEP in units

Step 1â€¦ 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.

Scenario 2 -5

Formula

#Units aboveBEP =NI/CM

Step 2â€¦ FindBEP in units

Step 1â€¦ 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

Scenario 2 -6

Formula

# Units belowBEP =(NI)/CM

Step 2â€¦ FindBEP in units

Step 1â€¦ 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

Scenario 2 -7

Formula

# units above BEP *CM = NI

Step 2â€¦ FindBEP in units

Step 1â€¦ 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

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 Marconiâ€™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

TotalVariable Costs were $150000

TotalFixed Costs were $180000

What was Marconiâ€™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 $

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 Marchoniâ€™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

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 Marchoniâ€™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

This completes Chapter 5