Math 141: Business Mathematics I
Fall 2015
§1.4 Intersection of Straight Lines
Instructor: Yeong-Chyuan Chung
Outline
• Finding the Point of Intersection of Two Straight Lines
• Applications of Finding Points of Intersection
– Break-Even Analysis / Decision Analysis
– Market Equilibrium
Finding Points of Intersection
Any two lines that are not parallel will intersect at one (and only one) point. If the two lines
represent some data, then the point of intersection may give us further information about
the data as we will see later. Thus, given two (non-parallel) lines, we need to know how to
find the point of intersection, both algebraically and using the graphing calculator (GC).
Example (Exercise 2 in the text). Find the point of intersection of the following lines
algebraically.
y = −4x − 7
−y = 5x + 10
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How to use the GC to find the point of intersection
1. Graph the two lines by pressing the Y= button and entering the slope-intercept form
of the equations into Y1 and Y2. Then press Graph.
2. If you don’t see the intersection in the screen, press Window to adjust the settings
(minimum and maximum values of x and y). You can also press Zoom and select
0:ZoomFit.
3. Press 2nd Trace and select 5:Intersect.
4. The calculator will ask you to select which two functions you are considering by asking
for the first curve and the second curve. The cursor will blink on a curve and show the
function name in the top left corner of the screen. You can press the up/down arrows
to change the selection. Since we only have two curves anyway, we can just hit Enter
twice without worrying about the selection.
5. The calculator will then ask you to guess where the intersection is located. Use the
left/right arrows to move the cursor close to the intersection. Then hit Enter.
6. The point of intersection will be displayed at the bottom of the screen.
Example. Use the GC to find the point of intersection of the lines in the previous example.
Break-Even Analysis / Decision Analysis
Consider a company with linear cost and revenue functions. Suppose we graph the two
functions and we get the following diagram:
What information about the company can we obtain from the diagram?
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The break-even point is the point at which the company neither makes a profit nor sustains
a loss. In mathematical language, it is the point of intersection of
Example (Exercise 12 in the text). A division of Carter Enterprises produces income tax
apps for smartphones. Each income tax app sells for $8. The monthly fixed costs incurred
by the division are $25,000, and the variable cost of producing each income tax app is $3.
(a) Find the cost function.
(b) Find the revenue function.
(c) Find the profit function.
(d) Find the break-even point algebraically.
(e) Find the break-even point using the graphing calculator.
(f ) How many units of the app must the company sell to make a profit of $10,000?
Suppose the division wants to produce and sell another app instead, and only one of the two
apps can be produced. They then have to make a decision, probably based on projected sales
figures and which option will give them a greater profit. Such a process is sometimes called
decision analysis. Other situations where decision analysis may be used include choosing
between two manufacturing processes for a product, or choosing between two job offers.
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Market Equilibrium
Under pure competition (a market with a broad range of competitors selling the same product), the price of a commodity eventually settles at a level dictated by the condition that the
supply equals the demand. When this happens, we say that market equilibrium prevails.
In mathematical language, it is the point of intersection of
The equilibrium quantity is
The equilibrium price is
Example (Exercise 28 in the text). The quantity demanded each month of Russo Espresso
Makers is 250 when the unit price is $140; the quantity demanded each month is 1000 when
the unit price is $110. The suppliers will market 750 espresso makers if the unit price is $60
or higher. At a unit price of $80, they are willing to market 2250 units. Both the demand
and supply equations are known to be linear.
(a) Find the demand equation.
(b) Find the supply equation.
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(c) Find the equilibrium quantity and the equilibrium price.
(d) Sketch the demand and supply curves, clearly indicating the coordinates of the point of
intersection.