Download Graphs of Linear Equations in Two Variables

Name __________________________________
Period __________
Date:
Essential Question: How does a Cartesian coordinate
system incorporate the concept of number lines?
Topic: 3-2 Graphs
of Linear
Equations in Two
Variables
Standard: A-REI.10
Understand that the graph of an equation in two variables is the set of all
its solutions plotted in the coordinate plane, often forming a curve (which
could be a line).
Objective:
To graph a linear equation in two variables.
Graphing:
Solutions of open sentences in x and y can be graphed in an
xy-coordinate plane. To set up a plane rectangular
coordinate system, draw two number lines, or axes, meeting at
right angles at a point O, the origin. The horizontal axis is
called the x-axis, and the vertical axis, the y-axis. The axes
divide the plane into four quadrants.
With each ordered pair of numbers, you can associate a unique
point in the plane. To associate a point with the pair (4, 3),
imagine drawing a vertical line through 4 on the x-axis and a
horizontal line through 3 on the y-axis, as shown below.
These lines intersect at a point P, the graph of (4, 3).
Locating a point in this way is called graphing the ordered
pair, or plotting the point.
Summary
Coordinates:
By reversing the process just described, you can associate with
each point P in the plane a unique ordered pair (a, b) of real
numbers, called the coordinates of P (see the diagram below).
The first coordinate, a. is called the x-coordinate, or abscissa,
of P; the second coordinate, b, is called the y-coordinate, or
ordinate of P.
This one-to-one correspondence between ordered pairs of real
numbers and points of the plane can be summarized as
follows:
1. There is exactly one point in the coordinate plane
associated with each ordered pair of real numbers.
2. There is exactly one ordered pair of real numbers
associated with each point in the coordinate plane.
A rectangular coordinate system is often called a Cartesian
coordinate system in honor of René Descartes (1596-1650),
the French philosopher and mathematician who introduced
coordinates.
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Example 1:
Graph the ordered pairs (8, 1), (0, 2), (2. 8), (2, 8), and
(4, 0) in the same coordinate plane.
(2, 8)
(0, 2)
(8, 1)
(4, 0)
(2, 8)
Exercise 1:
Graph the ordered pairs (5, 10), (0, 6), (2. 8), (8, 8), and
(4, 12) in the same coordinate plane.
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Example 2:
Solution:
Find and graph five solutions of
.
Solve for one variable, say y, in terms of the other.
Choose convenient values of x and find the corresponding
values of y. Then graph the resulting ordered pairs.
Exercise 2:
Find and graph five solutions of
.
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Graph:
The graph of an open sentence in two variables is the set of all
points in the coordinate plane whose coordinates satisfy the
open sentence. Example 2 suggests that if we were to graph all
the ordered pairs satisfying
, we would obtain the
line shown in blue. The following theorem, which is proved in
more advanced courses, tells us that the graph of the equation
is in fact a line.
Theorem
The graph of every equation of the form
(A and B not both 0)
is a line. Conversely, every line in the coordinate plane is
the graph of an equation of this form.
Because of this property, any equation that can be expressed in
the form
(A and B not both 0) is called a linear
equation in two variables. The equation
is
linear since it can be written as
.
The following equations are not linear:
Although you need only two points to determine the graph of a
linear equation, it is a good practice to plot a third point as a
check. Points where the graph crosses the axes are often easy
to find and plot.
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Example 3:
Solution:
Graph
.
The graph crosses the y-axis at a point whose x-coordinate is 0. The
graph crosses the x-axis at a point whose y-coordinate is 0.
Let
.
Let
( )
Solution (0, 3)
.
( )
Solution (
, 0)
The graph is the line through the points with coordinates (0, 3) and
( , 0).
As a check, note that (3, 5) is a solution of
graph lies on the line.
and its
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Exercise 3:
Graph
Example 4:
Graph in a coordinate plane.
.
a.
Solution:
The equation
written as
can be
The graph consists of all
points having y-coordinate
4 and is therefore a
horizontal line.
b.
Solution:
The equation
be written as
can
The graph consists of all
points having x-coordinate
and is therefore a
vertical line.
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Exercise 4:
Graph in a coordinate plane.
a.
b.
Example 4 illustrates the fact that the graph of Ax + By = C is
a horizontal line if A = 0, and is a vertical line if B = 0.
From now on, we will follow the common practice of using the
simpler phrase "the point (a, b)" instead of "the point whose
coordinates are (a, b)."
Also, the phrase "the line
whose equation is
Class work:
p 110 Oral Exercises: 1-31
Homework:
p 111 Written Exercises: 2-24 even
" will mean "the line
."
P 105 Written Exercises: 34-38 even
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