Applications of Linear Equations
-
Upload
cally-carlson -
Category
Documents
-
view
37 -
download
1
description
Transcript of Applications of Linear Equations
5 - 1Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
Applications of Linear EquationsApplications of Linear Equations
5 - 2Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 3Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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.
5 - 4Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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…
5 - 5Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 6Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
LO 2.LO 2.
5 - 7Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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?
5 - 8Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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…
5 - 9Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 10Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 11Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 12Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
AnalysisAnalysisCostCost
LO 3.LO 3.
5 - 13Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 14Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
Terminology
Break Even Point
… is the point at which
neither
a Profit or Loss is made
5 - 15Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 16Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
$ %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
5 - 17Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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?
5 - 18Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 19Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 20Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 21Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 22Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 23Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 24Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 25Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 26Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 27Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 28Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 29Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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?
5 - 30Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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 $
5 - 31Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 32Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
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
5 - 33Linear
Equations Apps.
Linear
Equations Apps.
McGraw-Hill Ryerson©
This completes Chapter 5This completes Chapter 5