MATH103Mathematics for Business and

Economics - IChapter 3 - Lines, Parabolas, and

Systems

Section 3-1 - Lines

y

xx2x1

y1

y2

Horizontal Change

Vertical Change

Many relationships between quantities can be represent conveniently by straight lines. The slope of a nonvertical line that passes through the points P(x1, y1) and Q(x2, y2) is denoted by m and is defined by

2 1

2 1

vertical change

Horizontal change

.

m

y y

x x

Q(x2, y2)

P(x1, y1)

Horizontal Change

Vertical Change

Vertical and Horizontal LinesEither the “rise” or “run” could be zero

m = 0

m is undefined

y

x

m (slope) is negative

m (slope) is positive

EXAMPLE 1 Finding and Interpreting the Slope of a Line

Sketch the graph of the line that passes through the points P(1, –1) and Q(3, 3). Find and interpret the slope of the line.

SolutionAny two points determine a line; the graph of the line passing through the points P(1, –1) and Q(3, 3) is sketched here.

Horizontal Change = 2

Vertical Change =4

EXAMPLE 1 Finding and Interpreting the Slope of a Line

P(1, –1) and Q(3, 3)

2 1

2 1change in -coordinate

change in -coordinates

3 1 42

3

s

3 1

1

2

3

1

y ym

x xx

y

Solution continued

The slope of this line is 2; this means that the value of y increases by exactly 2 units for every increase of 1 unit in the value of x. The graph is a straight line rising by 2 units for every one unit we go to the right.

Interpretation

EXAMPLE 2 Finding and Interpreting the Slope of a Line

Equations of LinesEquations of LinesPoint-Slope Form: To find the equation of a line, when you only have two points. The point-slope form of the equation of a line is

1 1( )y y m x x where m is the slope and (x1, y1) is a given point.

EXAMPLE 1 Finding an Equation of a Line with Given Point and Slope

Find the point–slope form of the equation of the line passing through the point (1, –2) and with slope m = 3. Then solve for y.

Solution

We have x1 = 1, y1 = –2, and m = 3.

y y1 m x x1 y 2 3 x 1 y 2 3x 3

y 3x 5

Find the equation of a line that passes through the point (1,-3) with slope of 2

EXAMPLE 2 Finding an Equation of a Line with Given Point and Slope

EXAMPLE 3 Finding an Equation of a Line Passing Through Two Given Points

Find the point–slope form of the equation of the line l passing through the points (–2, 1) and (3, 7). Then solve for y.

Solution

First, find the slope.m 7 1

3 2 6

3 2

6

5We have x1 = 3, y1 = 7.

y y1 m x x1 y 7

6

5x 3

y 7 6

5x

18

5

y 6

5x

17

5

19

9

)5(4

716

m

Now use the point-slope form with m = 1 and (x1, x2) = (4,16). (We could just as well have used (-5,7)).

Find the equation of the line through the points (-5,7) and (4,16).

Solution:

12164

)4(116

xxy

xy

EXAMPLE 4 Finding an Equation of a Line with two Given Points

EXAMPLE 5

Find the point–slope form of the equation of the line with slope m and y-intercept b. Then solve for y.

Finding an Equation of a Line with a Given Slope and y-intercept

Solution

The line passes through (0, b).

y y1 m x x1 y b m x 0 y b mx

y mx b

SLOPE–INTERCEPT FORM OF THEEQUATION OF A LINE

The slope-intercept form of the equation of the line with slope m and y-intercept b is

y mx b .

EXAMPLE 1 Slope – Intercept Form

EXAMPLE 2 Find the Slope and y-intercept of a line

EXAMPLE Equations of Horizontal and Vertical Lines

PARALLEL AND PERPENDICULAR LINES

Let l1 and l2 be two distinct lines with slopes m1 and m2,respectively. Thenl1 is parallel to l2 if and only if m1 = m2.l1 is perpendicular l2 to if and only if m1•m2 = –1.Any two vertical lines are parallel, and any horizontal line is perpendicular to any vertical line.

Section 3-2Applications and Linear Functions

EXAMPLE 1 Finding a Demand Equation

Note : The graph of a linear function can be drawn by using the same procedures as drawing a line.

EXAMPLE 2 Diet for Hens

Example

A business copier repair company charges a fixed amount plus an hourly rate for service. If a customer is billed $159 for a one-hour service call and $287 for a three hour service call, find the linear function that describes the price of a service call when x is the number of hours of service

We need to find the equation for a line of the form; y=mx+bWe know two points on the line: (1,159) and (3,287) So, m = (287 – 159)/(3 – 1) = 64Use point-slope approach to find the line◦ (y – y1) = 64(x – x1)

◦ y = 287 – 64(3) + 65x ◦ y = 95 + 64x

EXAMPLE 3

Example : Suppose consumer will demand 40 units of a Product when the price is $12 per unit and 25 units when The price is $18 each. Find the demand equation Assuming that is linear. Find the price per unit when 30 units are demanded

Solution: 

The Points are: = (40,12) , and =(25,18)

Slope Formula: x represent q and y represent p

Hence an equation of the line is

),( 11 yx ),( 22 yx

2 1

2 1

- 18 12 2

- 25 40 5

y ym

x x

212 40

52

285

p q

qp

EXAMPLE 4 Finding a Demand Equation

When q=30

285

2 qP

Q

P

28

70

2.30 28 12 28 16

5P

Graph of the demand equation p intercept q=0 then p=28, Q intercept p=0 then q=70

EXAMPLE 5 Finding a Demand Equation

Example Suppose a manufacturer of shoes will place on the market 50 (thousand pairs) when the price is 35 (dollars per pair) and 35 when the price is 30. Find the supply equation, assuming that price p and quantity q are linearly related

Solution:The Points are: =(50,35), and =(35,30)

Slope Formula: x represent q and y represent p

Hence the equation of the line is

2 1

2 1

30 35 1

35 50 3

p pm

q q

1 1 5535 50

3 3 3p q q

),( 11 yx ),( 22 yx

EXAMPLE 6 Finding a Supply Equation

EXAMPLE 7 Finding a Supply Equation

Example: Suppose the cost to produce 10 units of a product is

$40 and the cost of 20 units is $ 70. If cost c is linearly related to

output q, find a linear equation relating c and q. Find the cost to

produce 35 units.

Solution:The line passing through (10,40) and (20,70) has slope

So an equation for the line is:

If q=35 then c=3(35)+10=115

31020

4070

m

40 3 10

3 10

c q

q

EXAMPLE 8 Finding a Cost Equation

Office equipment was purchased for $20,000 and will have a scrap value of $2,000 after 10 years. If its value is depreciated linearly, find the linear equation that relates value (V) in dollars to time (t) in years:

Solution: When t = 0, V = 20,000 and when t = 10, V = 2,000. Thus, we have two ordered pairs (0, 20,000) and (10, 2000). We find the slope of the line using the slope formula. The y intercept is already knownThe slope is (2000-20,000)/(10 – 0) = -1,800. when t = 0, V = 20,000, so the y intercept is b=20,000Therefore, by slope-intercept form(y=mx+b) equation is V(t) = - 1,800t + 20,000.

Finding an EquationEXAMPLE 10