Download 1.4 Intersection of Straight Lines

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STRAIGHT LINES
AND LINEAR
FUNCTIONS
Warm Up: p. 40 Example 1
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1.4
Intersection of Straight Lines
Copyright © Cengage Learning. All rights reserved.
Example 1
Find the point of intersection of the straight lines that have
equations y = x + 1 and y = –2x + 4.
Solution:
We solve the given simultaneous equations. Substituting
the value y as given in the first equation into the second,
we obtain
x + 1 = –2x + 4
3x = 3
x=1
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Example 1 – Solution
cont’d
Substituting this value of x into either one of the given
equations yields y = 2. Therefore, the required point of
intersection is (1, 2) (Figure 36).
The point of intersection of L1 and L2 is (1, 2).
Figure 36
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Break-Even Analysis
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Break-Even Analysis
Consider a firm with (linear) cost function C(x), revenue
function R(x), and profit function P(x) given by
C(x) = cx + F
R(x) = sx
P(x) = R(x) – C(x) = (s – c)x – F
where c denotes the unit cost of production, s the selling
price per unit, F the fixed cost incurred by the firm, and
x the level of production and sales.
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Break-Even Analysis
The level of production at which the firm neither makes a
profit nor sustains a loss is called the break-even level of
operation and may be determined by solving the equations
y = C(x) and y = R(x) simultaneously.
At the level of production x0, the profit is zero, so
P(x0) = R(x0) – C(x0) = 0
R(x0) = C(x0)
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Break-Even Analysis
The point P0(x0, y0), the solution of the simultaneous
equations y = R(x) and y = C(x), is referred to as the
break-even point; the number x0 and the number y0 are
called the break-even quantity and the break-even
revenue, respectively.
Geometrically, the break-even point P0(x0, y0) is just the
point of intersection of the straight lines representing the
cost and revenue functions, respectively.
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Break-Even Analysis
This follows because P0(x0, y0), being the solution of the
simultaneous equations y = R(x) and y = C(x), must lie on
both these lines simultaneously (Figure 37).
P0 is the break-even point.
Figure 37
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Break-Even Analysis
Note that if x < x0, then R(x) < C(x), so
P(x) = R(x) – C(x) < 0;
thus, the firm sustains a loss at this level of production.
On the other hand, if x > x0, then P(x) > 0, and the firm
operates at a profitable level.
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Applied Example 4 – Decision Analysis
The management of Robertson Controls must decide
between two manufacturing processes for its model C
electronic thermostat.
The monthly cost of the first process is given by
C1(x) = 20x + 10,000 dollars, where x is the number of
thermostats produced; the monthly cost of the second
process is given by C2(x) = 10x + 30,000 dollars.
If the projected monthly sales are 800 thermostats at a unit
price of $40, which process should management choose in
order to maximize the company’s profit?
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Applied Example 4 – Solution
The break-even level of operation using the first process is
obtained by solving the equation
40x = 20x + 10,000
20x = 10,000
x = 500
giving an output of 500 units.
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Applied Example 4 – Solution
cont’d
Next, we solve the equation
40x = 10x + 30,000
30x = 30,000
x = 1000
giving an output of 1000 units for a break-even operation
using the second process. Since the projected sales are
800 units, we conclude that management should choose
the first process, which will give the firm a profit.
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Practice
p. 45 Self-Check Exercises #2a, b
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