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1.4 Intersections of Straight Lines
Suppose we are given two straight lines L1 and L2 with equations
y = m1x + b1 and y = m2x + b2
(where m1, b1, m2, and b2 are constants) that intersect at the point
P(x0, y0).
• The point P(x0, y0) lies on the line L1 and so satisfies the equation y
= m1x + b1.
• The point P(x0, y0) also lies on the line L2 and so satisfies
y = m2x + b2 as well.
• Therefore, to find the point of intersection P(x0, y0) of the lines L1
and L2, we solve for x and y the system composed of the two
equations
y = m1x + b1 and y = m2x + b2
Intersection Point of Two Lines
L1 : y  m1 x  b1
Given the two lines
L2 : y  m2 x  b2
m1 ,m2, b1, and b2 are constants
Find a point (x, y) that satisfies both equations.
y
L1
L2
x
Solve the system consisting of
y  m1 x  b1 and y  m2 x  b2
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Find the intersection point of the following pair of lines:
y  4x  7
y  2 x  17
Notice both are in
Slope-Intercept Form
4x  7  2x 17
6 x  24
x4
Substitute in for y
Solve for x
y  4x  7
 4(4)  7  9
Find y
Intersection point: (4, 9)
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
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) = c x + F
R (x) = s x
P (x) = R (x) – C (x)=(s - c) x - F
Where c denotes the unit cost of production, s denotes the selling
price per unit, F denotes the fixed cost incurred by the firm, and x
Denotes the level of production and sales.
The break-even level of operation is the level of production that
results in no profit and no loss. It may be determined by solving
p=C(x) and p=R(x) simultaneously.
For this level of production, the profit is zero, so
P( x0 )  R( x0 )  C ( x0 )  0
R( x0 )  C ( x0 )
The point P0 ( x0 , y0 ) , the solution of the simultaneous equations
P = R (x) and p = C (x), is referred to as the break-even point;
The number x0 and the number p0 are called the break-even
Quantity and the break-even revenue, respectively.
break-even
point
Dollars
Revenue
profit
loss
Cost
Units
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. A shirt producer has a fixed monthly cost of $3600. If each
shirt costs $3 and sells for $12, find the break-even point.
Let x be the number of shirts produced and sold
Cost: C(x) = 3x + 3600
Revenue: R(x) = 12x
Break even point: R( x)  C ( x)
12 x  3x  3600
x  400
R(400)  4800
At 400 units, the break-even revenue is $4800
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
A division of Career Enterprises produces “Personal Income Tax”
diaries. Each diary sells for $8. The monthly fixed costs incurred by
the division are $25,000, and the variable cost of producing each
diary is $3.
a. Find the break-even point for the division.
b. What should be the level of sales in order for the division to
realize a 15% profit over the cost of making the diaries?
a. R(x) = 8x; C(x) = 25,000 + 3x ,
P(x) = R(x) – C(x) = 5x – 25,000.
Next, the breakeven point occurs when P(x) = 0,
that is, 5x – 25,000 = 0
x = 5000.
Then R(5000) = 40,000,
so the breakeven point is (5000, 40,000).
b. If the division realizes a 15 percent profit over the cost of making
the diaries, then
P(x) = 0.15 C(x)
5x – 25,000 = 0.15(25,000 + 3x)
4.55x = 28,750
x = 6318.68
x  6319
Market Equilibrium
Market Equilibrium occurs when the quantity produced is equal
to the quantity demanded. The quantity produced at market
equilibrium is called the equilibrium quantity, and the
corresponding price is called the equilibrium price.
supply
price
curve
demand
curve
x units
Equilibrium Point
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex (optional). The maker of a plastic container has determined
that the demand for its product is 400 units if the unit price is $3
and 900 units if the unit price is $2.50. The manufacturer will
not supply any containers for less than or equal to $1 but for
each $0.30 increase in unit price above $1, the manufacturer will
market an additional 200 units. Both the supply and demand
functions are linear, find:
a. The demand function
b. The supply function
c. The equilibrium price and quantity
...
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Let p be the price in dollars and x be in units
a. The demand function
 x, p  :  400,3 and 900,2.5 ;
p  3  0.001 x  400
3  2.5
m
 0.001
400  900
p  0.001x  3.4
b. The supply function
 x, p  :  0,1 and  200,1.3;
m
1  1.3
 0.0015
0  200
p  0.0015 x  1
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
c. The equilibrium price and quantity
Solve p  0.001x  3.4
simultaneously.
and
p  0.0015 x  1
0.001x  3.4  0.0015x 1
0.0025x  2.4
x  960
p  0.0015(960)  1  2.44
The equilibrium quantity is 960 units at a price of $2.44 per
unit.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.