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:. Solve two linear equations with two variables. LO 1. LO 2. Solve problems that require setting up linear equations with two variables. - PowerPoint PPT PresentationTRANSCRIPT
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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
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
Solving Two Equations with Two Unknowns
Solving Two Equations with Two Unknowns
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
EquationEquation (2)
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 (1)
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
Step 1… Find CM Step 2… Find BEP in units
S = $185VC = - 148CM $ 37
= FC/CM= $4588/ $37= 124 Units
=BEPin units /PC*100
Step 3… Find% of Capacity
= 124/320*100= 39% of Capacity
VC =S*80% = $148
$148
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FC = $3136 VC = $157 S= $185 Capacity = 320 units
Scenario 2 -3
Determine the BEP as a % of capacity if S is reduced to $171.
= BEPin units /PC*100 FormulaFormula
Step 1… Find CM Step 2… Find BEP in units
S = $ 171VC = -157CM $ 14
= FC/CM = $3136/ $14= 224 Units
=BEPin units /PC*100
Step 3… Find% of Capacity
= 224/320*100= 70% of Capacity
$171
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FC = $3136 VC = $157 S= $185 Capacity = 320 units
Scenario 2 -4
Determine the NI if 134 units are sold!
Step 1… Find CM Step 2… Find BEP in units
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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FC = $3136 VC = $157 S= $185 Capacity = 320 units
Scenario 2 -5
What unit sales will generate NI of $2000?
#Units above BEP = NI/CM FormulaFormula
Step 1… Find CM Step 2… Find BEP in units
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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FC = $3136 VC = $157 S= $185 Capacity = 320 units
Scenario 2 -6
# Units below BEP = (NI)/CM FormulaFormula
Step 1… Find CM Step 2… Find BEP in units
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
Step 1… Find CM Step 2… Find BEP in units
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.
FC = $3136 VC = $157 S= $185 Capacity = 320 units
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