Download 1.3 - El Camino College

sec1.3 Lines in plane,Slope
Objective:: Review lines in a plane and slopes
and business applications
Slope intercept form of Equation of line
y=mx+b m = slope and y intercept =0, b
m=
change_in_y
y
y −y
=  x = x 22 − x 11 = rise
run
change_in_x
Vertical Line, slope undefined graph x = 2
Horizontal Line, slope = 0
graph y = 2
y
(black)
red
4
3
2
1
-4
-2
2
4
-1
x
-2
Linear equation y = 2x + 1
Linear equation y = −x + 2
y
positive slope
(black)
negative slope
(red)
4
2
-2
-1
1
2
3
4
5
x
-2
Linear equation x + y = 2
in slope intercept form solve for y,
y = −x + 2 the same graph as the red on above.
Example 2- Slope as ratio
Maximum recommemded slope of wheelchair ramp is 1/12 = 0. 083
A business store ramp rises 22 in. overall (height) and a horizontal length
of 24 in.
Is the ramp steeper than recommended?
—————————————
1
vertical_change
= 22in. = 0. 076
24in.
horizontal_change
0.076 ≺ 0. 083 so the slope is not steeper than recommended.
Solution: Slope m=
Example 3 -Using Slope as a Rate of Change
A manufacturing company determines the cost (C) in dollars of producing
x units of C = 25x + 3500
Describe the practical significance of the y intercept and slope.
——————————————–
Solution:
The y intercept is (0,3500). The fixed costs of producing
zero units is $3500, the cost paid regardless of the units produced
Slope m=$25 is the cost of producing each unit.
Economists call this the marginal costs
If production increases by one unit,the "margin " or extra cost is $25
C = 25x + 3500
y
8000
7000
6000
5000
4000
3000
2000
1000
0
0
10
20
30
40
50
60
70
80
90 100 110 120 130 140 150 160
x
Production Costs
————————————
Objective: Examine a Business application of Break-Even analysis
Include Sec 1.2 # 61 as part of your homework
Possible Test Question
Sec 1.2 Example 5 Finding a Break-Even Point (p49)
A business manufactures a product at a cost of $0.65 per unit and sells the product for
$1.20 per unit.
The company’s initial investment was $10,000 to produce the product (fixed costs)
a.Will the company break even if it sells 18,000 units?
b. How many units must the company sell to break even?
————————————————2
C = total cost of producing x units
R= revenue from the sale of x units
p = selling price per unit
C=mx+b where m = cost per unit of each item x
and b = initial investment and fixed costs
R=px where p = selling price per unit
The break even point is where Revenue=Costs
R=C
——————————–
a. Write the cost and revenue equations
C = 0. 65x + 10, 000
R = 1. 20x
Write the break even equation
R=C
1. 20x = 0. 65x + 10, 000
Substitute x=18,000 into the equation
to see if R=C
R
C
1. 20 ⋅ 18000
?
0. 65 ⋅ 18000 + 10000
21600
21700
Costs are greater than the Revenue so
the company will not break even
if it sells 18,000 units
b. What is the break even point?
Solve the break even equation for x
R=C
1. 20x = 0. 65x + 10, 000
Solve for x, the number of units
1. 20x − 0. 65x = 10000,
x = 18, 182 the number of units sold before
there is any profit
To find the revenue or costs at this value
evaluate
R = px
R = 1. 20x = 1. 20 ⋅ 18182 = $21818.
Break even point where R = C
is 18182, 21818
C= (black)
R= (red)
3
y
40000
30000
20000
10000
0
0
10000
20000
30000
x
Loss before the break even point is
C−R
costs are greater than sales
Profit after the break even point is
P = R−C
sales exceed costs
where P = profit
———————————–
Back to Sec 1.3
Finding the Slope of a Line
y
change_in_y
y −y
=  x = x 22 − x 11 = rise
slope m =
run
change_in_x
where point 1 =x 1 , y 1 
point 2 =x 2 , y 2 
Exampe 4-Finding the slope of a Line
Find the slope of the line passing through each pair of points
a. (-2,0) and (3,1)
y −y
1 − 0
m= x 22 − x 11 =
= 1 = 1
5
3+2
3 − −2
b. (-1,2) and (2,2)
y −y
m= x 22 − x 11 = 2 − 2 = 0 = 0
3+1
3 − −1
y
4
2
-2
2
4
x
-2
y = 2_horizontal_line
c. (0,4) and (1,-1)
y −y
m= x 22 − x 11 = −1 − 4 = −5 = −5
1−0
1
d. (3,4) and (3,1)
y −y
m= x 22 − x 11 = 1 − 4 = −3 = undefined
3−3
0
4
because division by zero is undefined
3, −3, 3, 5
y
4
2
-2
2
4
-2
x
x = 3_vertical_line
Point Slope form of the line
y − y 1 = mx − x 1 
Example 5-Using Point Slope form of line
Find the equation of the line that has slope 3 and passes thru the point (1,-2)
Answer in slope intercept form and standard form.
Solution:
y − y 1 = mx − x 1 
y − −2 = 3x − 1
y + 2 = 3x − 3
y = 3x − 5
slope_intercept_form
standard form ax + by = c
standard_form
3x − y = 5
Parallel (∥)and Perpendicular⊥ lines
Similar to Example 7 in text
Parallel Lines
Two lines are parallel (∥)if and only if they have the same slope or are both vertical.
Example Find an equation of the line through the point −3, −2 parallel to the line
2x − 7y + 6 = 0
Solution a. All you need to know is the slope of the line 2x − 7y + 6 = 0.
Solving for y gives
−7y = −2x − 6.
y = 27 x + 67
m = 27 (using the slope-intercept form of the line).
b. Find the equation (in point-slope form)
m= 27
Use y − y 1 = mx − x 1 
y − −2 = 27 x − −3
y + 2 = 27 x + 3
Multiplying both sides of the equation by 7
and using the distributive property gives
5
7y + 14 = 2x + 6
standard form
0 = 2x − 7y − 8
or
2x − 7y − 8 = 0
Note: Most texts use a positive x term first
Perpendicular Lines
Two lines with nonzero slopes m 1 and m 2 are perpendicular⊥ if and only if
m 1 m 2 = −1, that is, if their slopes are negative reciprocals (m 2 = − m11 ).
Also, any horizontal line (slope 0) is perpendicular to any vertical line (slope undefined).
Example Find an equation of the line through the point −3, −2 perpendicular to the line
2x − 7y + 6 = 0
Solution
a. Find the slope of the given line:
Example 2x − 7y + 6 = 0
gives ,as above, m=
So the slope of the given line is
2
7
2
7
.
The slope of a line perpendicular to this is then the negative reciprocal
m 2 = − m11
− 12 = − 72
7
b. You want the equation of a line through −3, −2 with a slope of − 72 :
Use y − y 1 = mx − x 1 
y − −2 = − 72 x − −3
y + 2 = − 72 x + 3
Multiplying both sides of the equation by −2
and using the distributive property gives
− 2y − 4 = 7x + 21
0 = 7x + 2y + 25
or
7x + 2y + 25 = 0
2
6
original y = 7 x + 7 blue
parallel
2x − 7y − 8 = 0black
perpendicular
2x − 7y + 6 = 0.
y = − 72 x − 252 red
6
y
4
2
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-2
Example 8 - Depreciating equipment
A ocmpany purchased a $12,000 machine with a useful life of 8 yrs.
The salvage value is then $2000.
a. Write a linear equation that describes the book value of the machine each year.
b. Show a table for each year using this equation
.——————————–
Solution:
Use V = mt + b where V = value of the machine at any time t
m = depreciation rate in dollars per year
b= value at t = 0
t
0
8
V 12,000
2000
m= V 2 − V 1 = 2000 − 12000 = −1250
t2 − t1
8−0
depreciation rate in dollars/yr. noted by the negative slope
Using slope intercept equation
V=mt+b
V = −1250t + 12000
y
12000
10000
8000
6000
4000
2000
0
0
2
4
6
8
x
b. V = −1250t + 12000
7
t
0
1
2
3
4
5
6
7
8
V 12,000 10750 9500 8250 7000 5750 4500 3250 2000
8