Applications of Linear Equations

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Applications of Linear Equations
Chapter 5
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After completing this chapter, you will be able
to
Solve two linear equations with
two variables
LO 1.
Solve problems that require setting up linear
equations with two variables
LO 2.
Also
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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.
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Solving Two Equations with Two Unknowns
LO 1.
2x 3y 6 x y 2
Equations
(A) Solve for y
(B) Solve for x
2x 3y 6 x y 4
(A) Solve for y
2x 2y 4
- 5y -10
y 2
(B) Solve for x
2x 3y 6
2x 3(2) 6
2x 6 6
2x 6 6
Check
x 0
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Solving Two Equations with Two Unknowns
You should always check your answer by
substituting the values into each of the
equations!
x 0 y 2
Equation 1
Equation 2
Left Side
Right Side
Left Side
Right Side
2x 3y
x y
2(0) 3(2)
0 2
6
2
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Setting up linear equations with Two Variables
LO 2.
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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?
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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.
1.15x
1.14y
91.70
(2)
the weekly bill rose from 84.40 to
91.70.
C.
Solving
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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
continue
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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
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Litres of Milk
50
1.15
57.50
Cans of Orange Juice
30
1.14
34.20
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LO 3.
Cost
Analysis
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do NOT change if sales
increase or decrease e.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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is the point at which neither
a Profit or Loss
is made
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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
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A Contribution Margin Statement
Net Sales(Price Units Sold)
x 100 Less Variable
Costs x x
Contribution Margin x x
Less Fixed Costs
x x
Net Income x x
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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
How much does the sale of an additional unit of a
firms product contribute towards increasing its
net income?
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Formulae
- To Find -
CM S - VC
Contribution Margin
Contribution Rate
CR CM/S 100
Break Even Point
x FC / CM
...in Units (x)
x (FC / CM) S
...in Sales
...in of Capacity
BEPin Units/PC100
At Break Even, Net Profit or Loss 0
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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.
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
432/ 900100 48 of Capacity
BEPin units PC100
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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

• Calculate the breakeven point (BEP)
• in units
• in dollars
• as a percent of capacity

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FC 3136 VC 157. S 185 Capacity 320
units
Break Even Point
in units
FC / CM
3136/
28
112 Units
in dollars
(FC / CM) S
(3136/
28)
185 20720
as a percent of capacity
BEPin units/PC100
112/320 100 35 of Capacity
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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/PC100
Step 3 Find of Capacity
Step 2 Find BEP in units
Step 1 Find CM
BEPin units /PC100
S 185
FC/CM
VC - 157 CM 28
2688/ 28
96/320100
96 Units
30 of Capacity

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FC 3136 VC 157 S 185 Capacity 320
units
148
VC S80 148
Determine the BEP as a of capacity if
FC are increased to 4588, and VC
reduced to 80 of S.
Step 3 Find of Capacity
Step 2 Find BEP in units
Step 1 Find CM
BEPin units /PC100
S 185
FC/CM
VC - 148 CM 37
4588/ 37
124/320100
124 Units
39 of Capacity
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FC 3136 VC 157 S 185 Capacity 320
units
171
Determine the BEP as a of capacity if
S is reduced to 171.
Step 3 Find of Capacity
Step 2 Find BEP in units
Step 1 Find CM
BEPin units /PC100
S 171
FC/CM
VC -157 CM 14
3136/ 14
224/320100
224 Units
70 of Capacity
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FC 3136 VC 157 S 185 Capacity 320
units
Determine the NI if 134 units are sold!
Units Sold 134 BEP
112 Over BEP 22
Step 2 Find BEP in units
Step 1 Find CM
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.
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FC 3136 VC 157 S 185 Capacity
320 units
What unit sales will generate NI of 2000?
NI/CM
2000/28 per Unit
72 Units above Break Even
CM of 28 per unit
72 Units 112 BEP Units Total Sales Units
184
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FC 3136 VC 157 S 185 Capacity 320
units
What are the unit sales if there is a Net Loss of
336?
(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
FC 3136 VC 157 S 185 Capacity 320
units
272
The company operates at 85 capacity. Find the
Profit or Loss.
320.85 272
Units Production 272 BEP 112
Over BEP 160
CM of 28 per unit
160 Units 28 Profit 4480
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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 Marconis BEP expressed in dollars of sales?
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What information is needed to calculate the BEP?
2. VC per Unit
1. Number of Units sold
3. CM
4. Total Costs
5. BEP in
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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. VC per Unit
Total VC Total Unit Sales
0.40pu

S 1.00 VC .40 CM .60
3. CM
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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
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