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

2. Solve two linear equations with two variables
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
Also

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

5. Solving Two Equations with Two Unknowns
2x – 3y = – 6
x + y = 2
Equations
Note
You should always check your answer by substituting the values into each of the equations!
x = 0 y = 2
Left Side
Right Side =
Show
Equation 1
Equation 2
Left Side
Right Side
Left Side
Right Side
= 2x – 3y
= – 6
= 2
= x + y
= 2(0) – 3(2)
Substituting
= 0 + 2
= – 6
= 2
LS = RS
LS = RS

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

7. York Daycare purchases the same amount of
Setting up
linear equations
with Two Variables Q
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?

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

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

10. Setting up linear equations with Two Variables Proof
(1)
x y = 76.73
x y = 79.74
Equation
(2)
.1004y =
Subtract
y = i.e. 30 cans
1.10x y = 84.40
Equation
(1)
Substitute into
1.10x (29.98) = 84.40
1.10x = 84.40
1.10x =
1.10x = 55.02
x = i.e. 50 litres
Proof

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

12. LO 3.
Cost
Analysis

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

14. … is the point at which neither a Profit or Loss is made
Terminology
Break Even Point
… is the point at which neither a Profit or Loss is made

15. …is the dollar amount expressed as a percent (%) of Net Sales
Terminology
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
A Contribution Margin statement

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

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

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

19. Applying the Formulae CM = S - VC = $50 - $30 = $20 CR = CM/S * 100%
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.
CM = S - VC
= $50 - $30 = $20
CR = CM/S * 100%
= $20/$50 * 100 = 40%
Break Even Point:
Units x = FC / CM
= $8640/$20 = 432 Units
In $ x = (FC / CM)* S
= ($8640/$20)* $50 = $21,600
BEPin units PC*100
= 432/ 900*100 = 48% of Capacity

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

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

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

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

25. Determine the NI if 134 units are sold!
Scenario 2 -4
FC = $3136 VC = $157 S= $185 Capacity = 320 units
Determine the NI if 134 units are sold!
NI = #Units above BEP*CM
Formula
Step 2… Find BEP in units
Units
Sold
BEP
Over BEP
Step 1… Find CM
S = $185
= FC/CM
VC =
CM $ 28
= $3136/$28
= 112 Units
CM of $28 per unit
Company had a NI of 22
* $28 = $616.

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

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

28. The company operates at 85% capacity. Find the Profit or Loss.
Scenario 2 -7
FC = $3136 VC = $157 S= $185 Capacity = 320 units
272
The company operates at 85% capacity. Find the Profit or Loss.
320*.85
= 272
Step 1… Find CM
Step 2… Find BEP in units
S = $185
VC =
CM $ 28
= FC/CM
= $3136/$28
= 112 Units
Units
Production 272
BEP
Over BEP
CM of $28 per unit
# units above BEP *CM = NI
Formula
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
Total Variable Costs were $150000
Total Fixed 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
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?
1. Number of Units sold
2. VC per Unit
3. CM
4. Total Costs
5. BEP in $

31. Case = = $0.40pu 1. Number of Units sold
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?
1. Number of Units sold
Let S = $1 and X be the Number of $1 Units sold
Sales of $ = Total Units sold
2. VC per Unit
Total VC
Total Unit Sales
$150000
375000 =
= $0.40pu
S $1.00 VC
CM $ .60
3. CM

32. Case $BEP = (FC/CM)*S = ($180000/0.60)*$1.00 = (300000)*$1.00
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?
4. Total Costs
TC = FC + VC
= $ X
$BEP = (FC/CM)*S
5. BEP in $
= ($180000/0.60)*$1.00
= (300000)*$1.00
# Of Units
= $300000
$BEP

33. This completes Chapter 5