Download Section P.4 Linear Equations in Two Variables Important Vocabulary

Section P.4
13
Linear Equations in Two Variables
Course Number
Section P.4 Linear Equations in Two Variables
Instructor
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
Date
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 38−40)
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 ).
To graph the line y = mx + b on the coordinate plane, . . .
plot
5
the y-intercept on the y-axis of the coordinate plane and use the
slope to move to another point on the line by rising (or falling) m
3
units for each unit of horizontal change from left to right.
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.
1
x
-5
-3
-1
-1
-3
-5
The equation of a horizontal line is
horizontal line is
0
y=b
. The slope of a
. To graph a horizontal line, . . .
draw a horizontal line through the y-intercept.
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
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y
1
3
5
14
Chapter P
Prerequisites
The y-coordinate of every point on the graph of a horizontal line
is
.
b
The equation of a vertical line is
vertical line is
undefined
x=a
. The slope of a
. To graph a vertical line, . . .
draw a vertical line through the value a in the equation x = a.
The x-coordinate of every point on the graph of a vertical line
is
.
a
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
In real-life problems, the slope of a line can be interpreted as
either a(n)
or a(n)
ratio
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
.
II. Finding the Slope of a Line (Pages 41−42)
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.
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
Section P.4
15
Linear Equations in Two Variables
III. Writing Linear Equations in Two Variables
(Pages 43−44)
The point-slope form of the equation of a line is
y − y1 = m(x − x1)
.
The two-point form of the equation of a line is
y − y1 = (y2 − y1)/(x2 − x1)⋅(x − x1)
.
The general form of the equation of a line is
Ax + By + C = 0
.
All equations of lines can be written in general form.
Which form of the equation of a line is most convenient when
given:
(a) the slope m and the y-intercept (0, b)?
slope-intercept form
(b) the slope m and a point (x1, y1) on the graph of the line?
point-slope form
(c) two points (x1, y1) and (x2, y2) that are on the graph of the
line?
two-point form
For the conditions in (a), (b), and (c) above, is it possible to use
only the slope-intercept form to find an equation? Explain.
Yes, unless the line is vertical. In cases (b) and (c),
it may be necessary to solve for the y-intercept b.
In case (c), the slope must be found from the given
points first.
Is it possible to use only the point-slope form to find an
equation? Explain.
Yes, unless the line is vertical. In case (c), the
slope must be found from the given points first.
Example 3: Find an equation of the line that passes through the
points (1, 5) and (− 3, 7) using (a) the slopeintercept form and (b) the point-slope form.
(a) y = − 0.5x + 5.5
(b) y − 5 = − 0.5(x − 1)
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to write linear
equations in two
variables
16
Chapter P
IV. Parallel and Perpendicular Lines (Pages 45−46)
Two lines are
parallel
if they do not intersect.
Two lines are
perpendicular
if they intersect at right
Prerequisites
What you should learn
How to use slope to
identify parallel and
perpendicular lines
angles.
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 40 and 46)
What you should learn
How to use linear
equations in two
variables to model and
solve real-life problems
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.
Additional notes
5
-5
-3
y
5
y
5
3
3
3
1
1
1
-1
-1
1
3
5
x
-5
-3
-1
-1
1
3
5
x
-5
-3
-1
-1
-3
-3
-3
-5
-5
-5
y
1
3
5
x
Homework Assignment
Page(s)
Exercises
Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE
Copyright © Houghton Mifflin Company. All rights reserved.