Download Section 1.3 Linear Equations in Two Variables

Section 1.3
•
Linear Equations in Two Variables
9
Name______________________________________________
Section 1.3 Linear Equations in Two Variables
Objective: In this lesson you learned how to find and use the slopes of
lines to write and graph linear equations in two variables.
Important Vocabulary
Define each term or concept.
Slope The number of units a nonvertical line rises (or falls) vertically for each unit of
horizontal change from left to right.
Parallel Two distinct nonvertical lines are parallel if and only if their slopes are
equal. That is, m1 = m2.
Perpendicular Two nonvertical lines are perpendicular if and only if their slopes are
negative reciprocals of each other. That is, m1 = − 1/m2.
I. Using Slope (Pages 25−26)
What you should learn
How to use slope to
graph linear equations in
two variables
The equation y = mx + b is called a linear equation in two
variables because . . .
the graph of this equation in the two
variables x and y is a straight line.
A line whose slope is positive
rises
from left to right.
A line whose slope is negative
falls
from left to right.
The slope-intercept form of the equation of a line is
y = mx + b
, where m is the
slope
and the
y-intercept is ( 0 , b ).
A vertical line has an equation of the form
x=a
. The
equation of a vertical line cannot be written in the form
y = mx + b because . . .
the slope of a vertical line is
y
5
undefined.
Example 1: Explain how to graph the linear equation
y = − 2/3x − 4. Then sketch its graph.
Because b = − 4, the y-intercept is (0, − 4).
Because the slope is − 2/3, the line falls 2 units
for every 3 units the line moves to the right.
3
1
x
-5
-3
-1
-1
-3
-5
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
1
3
5
10
•
Chapter 1
Functions and Their Graphs
Example 2: Sketch and label the graph of (a) y = − 1 and
(b) x = 3.
y
y
(a)
(b)
5
5
-5
-3
3
3
1
1
-1-1
1
3
5
x
-5
-3
-1-1
-3
-3
-5
-5
1
3
5
x
II. Finding the Slope of a Line (Pages 27−28)
The formula for the slope of a line passing through the points
(x1, y1) and (x2, y2) is m =
(y2 − y1)/(x2 − x1)
What you should learn
How to find slopes of
lines
.
To find the slope of the line through the points (− 2, 5) and
(4, − 3), . . .
subtract − 3 from 5 and divide this result by the
difference of − 2 and 4.
If a line falls from left to right, it has
line is horizontal, it has
vertical, it has
right, it has
undefined
positive
zero
negative
slope. If a
slope. If a line is
slope. If a line rises from left to
slope.
III. Writing Linear Equations in Two Variables (Page 29)
The point-slope form of the equation of a line is
y − y1 = m(x − x1)
.
The point-slope form is most useful for . . .
What you should learn
How to write linear
equations in two
variables
finding the
equation of a line.
The two-point form of the equation of a line is
y − y1 = (y2 − y1)/(x2 − x1)⋅(x − x1)
.
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
Section 1.3
•
Linear Equations in Two Variables
Name______________________________________________
Example 3: Find an equation of the line that passes through the
points (1, 5) and (− 3, 7) using (a) the point-slope
form and (b) the slope-intercept form.
(a) y − 5 = − 0.5(x − 1)
(b) y = − 0.5x + 5.5
IV. Parallel and Perpendicular Lines (Page 30)
What you should learn
How to use slope to
identify parallel and
perpendicular lines
The relationship between the slopes of two lines that are parallel
is . . .
that the slopes are equal.
The relationship between the slopes of two lines that are
perpendicular is . . .
that the slopes are negative reciprocals of
each other.
A line that is parallel to a line whose slope is 2 has slope
2
.
A line that is perpendicular to a line whose slope is 2 has slope
− 1/2 .
V. Applications of Slope (Pages 31−33)
In real-life problems, the slope of a line can be interpreted as
either a(n)
ratio
or a(n)
rate
. If the x-axis
and y-axis have the same unit of measure, then the slope has no
units and is a
ratio
. If the x-axis and y-axis have
different units of measure, then the slope is a
change
rate or rate of
.
Describe a real-life situation in which slope is a ratio.
Answers will vary.
Describe a real-life situation in which slope is a rate of change.
Answers will vary.
Linear or straight-line depreciation is . . .
used when the same
amount of depreciation is claimed each year over the useful life
of property.
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to use linear
equations in two
variables to model and
solve real-life problems
11
12
Chapter 1
•
Functions and Their Graphs
The prediction method in which an estimated point does not lie
between the given points is called
.
linear extrapolation
When the estimated point lies between two given points, the
procedure is called
linear interpolation
.
Every line has an equation that can be written in general form,
which is given as
, where A
Ax + By + C = 0
and B are not both zero.
Additional notes
y
y
5
5
3
3
3
1
-5
-3
y
5
-1-1
1
1
3
5
x
-5
-3
-1-1
1
1
3
5
x
-5
-3
-1-1
-3
-3
-3
-5
-5
-5
1
3
5
x
Homework Assignment
Page(s)
Exercises
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.