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 intersect at the point P(x 0, y 0 ). The point P(x 0, y 0 ) lies on the line L 1 and so satisfies the equation y = m 1 x + b 1. The point P(x 0, y 0 ) also lies on the line L 2 and so satisfies y = m 2 x + b 2 as well. Therefore, to find the point of intersection P(x 0, y 0 ) of the lines L 1 and L 2, we solve for x and y the system composed of the two equations y = m 1 x + b 1 and y = m 2 x + b Intersections of Straight Lines

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Intersection Point of Two Lines Given the two lines m 1,m 2, b 1, and b 2 are constants Find a point (x, y) that satisfies both equations. Solve the system consisting of L1L1 L2L2 x y

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Find the intersection point of the following pair of lines: Notice both are in Slope-Intercept Form Substitute in for y Solve for x Find y 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 + 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.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. For this level of production, the profit is zero, so Dollars Units loss Revenue Cost profit break-even point The point, 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.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Cost: C(x) = 3x 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 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, x, 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 = 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, x) 4.55x = 28,750 x =

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 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. price x units supply curve demand curve 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. a. The demand function b. The supply function Let p be the price in dollars and x be in units

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. c. The equilibrium price and quantity Solveand simultaneously. The equilibrium quantity is 960 units at a price of $2.44 per unit.