Equations of Lines - §3.4
Fall 2013 - Math 1010
y = mx + b
(y − y1 ) = m(x − x1 )
Ax + By = C
(Math 1010)
M 1010 §3.4
1 / 11
Roadmap
I
Discussion/Activity: Graphs and linear equations.
I
Form: The Point-Slope Equation
I
Form: Vertical, Horizontal, Parallel, and Perpendicular Lines
I
Applications
I
Discussion on homework, quizzes, and exams.
(Math 1010)
M 1010 §3.4
2 / 11
Point-Slope
Recall the slope formula re-imagined without fractions.
Slope:
(y2 − y1 ) = m(x2 − x1 )
This formula becomes the point-slope equation of a line when a slope, m,
is known along with only one point, (x1 , y1 ).
Point-slope form:
(y − y1 ) = m(x − x1 )
(Math 1010)
M 1010 §3.4
3 / 11
Example - Point-Slope
Write an equation of the line passing through the point (2, −7) with a
slope of m = 4.
(Math 1010)
M 1010 §3.4
4 / 11
Example - Point-Slope
Write an equation of the line passing through the point (2, −7) with a
slope of m = 4.
y − (−7) = 4(x − 2)
(Math 1010)
M 1010 §3.4
4 / 11
Example - Point-Slope
Write an equation of the line passing through the point (2, −7) with a
slope of m = 4.
y − (−7) = 4(x − 2)
y + 7 = 4(x − 2)
(Math 1010)
M 1010 §3.4
4 / 11
Example - Point-Slope
Slope-intercept forms y = mx + b pass through the point (0, b). Then the
point-slope form looks like:
y − b = m(x − 0)
.
(Math 1010)
M 1010 §3.4
5 / 11
Example - Point-Slope
Slope-intercept forms y = mx + b pass through the point (0, b). Then the
point-slope form looks like:
y − b = m(x − 0)
.
Write the point-slope form of the line through (−2, 1) and (4, 2), then
write its slope-intercept form.
(Math 1010)
M 1010 §3.4
5 / 11
Example - Point-Slope
Slope-intercept forms y = mx + b pass through the point (0, b). Then the
point-slope form looks like:
y − b = m(x − 0)
.
Write the point-slope form of the line through (−2, 1) and (4, 2), then
write its slope-intercept form.
m=
2−1
1
=
4 − (−2)
6
1
y − 1 = (x + 2)
6
1
4
y= x+
6
3
(Math 1010)
M 1010 §3.4
5 / 11
Special Forms
. Each point has
Horizontal lines have a slope of
y -coordinate b, from its
Vertical lines have an
a, from its
(0, b).
slope. Each point has x-coordinate
(a, 0).
Euclid formulated geometric axioms, one of which is that there is only one
line through a given point that is parallel to another line. Recall that
parallel lines have equal slopes. Perpendicular lines have
opposite-and-reciprocal slopes.
(Math 1010)
M 1010 §3.4
6 / 11
Special Forms
. Each point has
Horizontal lines have a slope of
y -coordinate b, from its
Vertical lines have an
a, from its
(0, b).
slope. Each point has x-coordinate
(a, 0).
Euclid formulated geometric axioms, one of which is that there is only one
line through a given point that is parallel to another line. Recall that
parallel lines have equal slopes. Perpendicular lines have
opposite-and-reciprocal slopes.
Blanks: zero, y -intercept, undefined, x-intercept
(Math 1010)
M 1010 §3.4
6 / 11
Summary of Forms of Equations of Lines
Algebraic Form
y = mx + b
Slope-Intercept
(y − y1 ) = m(x − x1 )
Point-Slope
Ax + By = C
x =a
Standard Form
Vertical line
y =b
Horizontal line
m1 = m2
m1 = − m12
(Math 1010)
Name of the Form
Parallel lines
Perpendicular lines
M 1010 §3.4
7 / 11
Application - Depreciation
The value of a car decreases in terms of time t. Let’s assume this to be
linear depeciation.
Set-up: The car’s initial value is $38,000. After 7 years it will be valued at
$7,000.
Write an equation for the straight-line depreciation of the value of the car.
(Math 1010)
M 1010 §3.4
8 / 11
Application - Depreciation
The value of a car decreases in terms of time t. Let’s assume this to be
linear depeciation.
Set-up: The car’s initial value is $38,000. After 7 years it will be valued at
$7,000.
Write an equation for the straight-line depreciation of the value of the car.
Use the equation to find the value of the car 2 years from its initial value.
(Math 1010)
M 1010 §3.4
8 / 11
Application - Depreciation
The value of a car decreases in terms of time t. Let’s assume this to be
linear depeciation.
Set-up: The car’s initial value is $38,000. After 7 years it will be valued at
$7,000.
Write an equation for the straight-line depreciation of the value of the car.
Use the equation to find the value of the car 2 years from its initial value.
Graph the equation. When does the value of the car become $0?
(Math 1010)
M 1010 §3.4
8 / 11
Application - Cost
The total cost to produce x items combines the overhead cost and cost to
produce one unit.
Set-up: To make hats, the total cost is the sum of the overhead of $20
and unit cost of $6 per item.
Write an equation for the total cost of producing x hats.
(Math 1010)
M 1010 §3.4
9 / 11
Application - Cost
The total cost to produce x items combines the overhead cost and cost to
produce one unit.
Set-up: To make hats, the total cost is the sum of the overhead of $20
and unit cost of $6 per item.
Write an equation for the total cost of producing x hats.
Use the equation to find the cost of make 40 products.
(Math 1010)
M 1010 §3.4
9 / 11
Application - Cost
The total cost to produce x items combines the overhead cost and cost to
produce one unit.
Set-up: To make hats, the total cost is the sum of the overhead of $20
and unit cost of $6 per item.
Write an equation for the total cost of producing x hats.
Use the equation to find the cost of make 40 products.
A budget constraint of $300 is introduced. Use either the equation or its
graph to estimate how many hats can be produced under this constraint.
(Math 1010)
M 1010 §3.4
9 / 11
Application - Demand
Demand relates the price p of a service and the demand d at that price.
This relationship may be linear.
Set-up: From 2010, raffle tickets priced at $4 sold 2000 tickets. From
2011, raffle tickets priced at $5 sold 1800 tickets.
Write a linear equation for the demand of tickets sold priced at p dollars.
(Math 1010)
M 1010 §3.4
10 / 11
Application - Demand
Demand relates the price p of a service and the demand d at that price.
This relationship may be linear.
Set-up: From 2010, raffle tickets priced at $4 sold 2000 tickets. From
2011, raffle tickets priced at $5 sold 1800 tickets.
Write a linear equation for the demand of tickets sold priced at p dollars.
Use the equation to find the demand of tickets sold at $10 per ticket.
(Math 1010)
M 1010 §3.4
10 / 11
Application - Demand
Demand relates the price p of a service and the demand d at that price.
This relationship may be linear.
Set-up: From 2010, raffle tickets priced at $4 sold 2000 tickets. From
2011, raffle tickets priced at $5 sold 1800 tickets.
Write a linear equation for the demand of tickets sold priced at p dollars.
Use the equation to find the demand of tickets sold at $10 per ticket.
Use the equation to find the demand of tickets sold at $2 per ticket.
(Math 1010)
M 1010 §3.4
10 / 11
Assignment
Assignment:
For Wednesday:
1. Exercises from §3.4 due Wednesday, September 25.
2. Quiz # 3: Graphs, Linear Equations
3. Read section 3.6. (Skip 3.5)
(Math 1010)
M 1010 §3.4
11 / 11