Suppose we are given two straight lines L 1 and L 2 with equations y = m 1 x + b 1 and y = m 2 x + b 2 (where m 1, b 1, m 2, and b 2 are constants) that

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

1.4 Intersections of Straight Lines

Copyright © 2006 Brooks/Cole, a division of Thomson Learning,

Inc.

Intersection Point of Two Lines

Given the two lines 1 1 1

2 2 2

:

:

L y m x b

L y m x b

m1 ,m2, b1, and b2 are constants

Find a point (x, y) that satisfies both equations.

Solve the system consisting of

L1

L2

1 1 2 2 and y m x b y m x b

x

y


Inc.

Ex. Find the intersection point of the following pair of lines:

4 7

2 17

y x

y x

Notice both are in Slope-Intercept Form

4 7 2 17x x Substitute in for y

6 24

4

x

x

Solve for x

Find y4 7

4(4) 7 9

y x

Intersection point: (4, 9)

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 + FR (x) = s xP (x) = R (x) – C (x)=(s - c) x - F

Where c denotes the unit cost of production, s denotes the sellingprice per unit, F denotes the fixed cost incurred by the firm, and xDenotes 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.


Inc.

For this level of production, the profit is zero, so

Dollars

Units

loss

Revenue

Cost

profit

break-even point

0 0 0

0 0

( ) ( ) ( ) 0

( ) ( )

P x R x C x

R x C x

The point , the solution of the simultaneous equationsP = 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.

0 0 0( , )P x y


Inc.

Cost: C(x) = 3x + 3600

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

Revenue: R(x) = 12x

Break even point: ( ) ( )

12 3 3600

400

R x C x

x x

x

(400) 4800R

At 400 units, the break-even revenue is $4800

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 6319x


Inc.

Market EquilibriumMarket 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.

price

x units

supply curve

demand curve

Equilibrium Point


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 . . .


Inc.

a. The demand function

, : 400,3 and 900,2.5 ;x p 3 2.50.001

400 900m

3 0.001 400p x

0.001 3.4p x

b. The supply function

, : 0,1 and 200,1.3 ;x p1 1.3

0.00150 200

m

0.0015 1p x

Let p be the price in dollars and x be in units


Inc.

c. The equilibrium price and quantity

Solve 0.001 3.4p x 0.0015 1p x and

simultaneously.

0.001 3.4 0.0015 1x x 0.0025 2.4

960

x

x

The equilibrium quantity is 960 units at a price of $2.44 per unit.

0.0015(960) 1 2.44p

Documents

Suppose we are given two straight lines L 1 and L 2 with equations y = m 1 x + b 1 and y = m 2 x + b 2 (where m 1, b 1, m 2, and b 2 are constants) that