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Graphing Linear Equations and Inequalities

5.1 The Rectangular Coordinate System
The rectangular coordinate system (figure 1)
has four quadrants created by intersecting a
horizontal and a vertical number line . The
horizontal number line is called the horizontal
axis, often labeled the x-axis. The vertical
number line is called the vertical axis, often
labeled the y-axis. When the axes intersect at
their zero points , the point of intersection is
called the origin. The quadrants are labeled in a
counterclockwise direction with Quadrant I in
the upper-right, Quadrant II in the upper-left,
Quadrant III in the lower-left, and Quadrant
in the lower-right.

Figure 1 The rectangular
coordinate system.

The location of every point in the
rectangular coordinate system is identified
by a pair of numbers called an ordered
(x, y). The first comp onent of an
pair is called the abscissa (often
referred to as the x-coordinate); it is the
horizontal distance from the point to the
y-axis. The second component of an
ordered pair is called the ordinate (often
referred to as the y-coordinate); it is the
vertical distance from the point to the x-axis.
The graph or plot of a point in the
plane is a dot placed at the location given
by the ordered pair of the point. For
example, to graph the ordered pair (4, 2)
place a dot 4 units to the right and 2 units
up from the origin (0, 0). See figure 2. Try
graphing the point (2, 4) in figure 2.

Figure 2 To graph the ordered pair (4, 2),
place a dot 4 units to the right of and 2 units
up from the origin, (0, 0).

Example 1
Graph the fol lowing ordered pairs. Label each point and
identify the quadrant each
point lies in.

1. A(2, 5) and B(5, 2)

2. C(3, -6) and D(-6, 3)

3. E(4, 0) and F(0, 4)

4. G(-5, -3) and H(-3, -5)

Example 2
Identify the ordered-pair location of each point on the

5.1.2 Scatter Diagrams

A scatter diagram is the graph of known ordered-pair data. A
scatter diagram illustrates the visual relationship between the two
variables in the ordered pair.

Example 2
A study by the Federal Aviation Administ ration showed that narrow,
over-the-wing emergency exit rows slow passenger evacuation. The
table below shows the space between seats, in inches, and the
evacuation time for 35 passengers. Graph the scatter diagram for the

x y

Space Between Seats (inches)

5.1.3 Average Rate of Change
Given two ordered-pairs and the average rate of
(ARC) of y with respect to x is given by

Example 3
From the data in example 3, find the average rate of change (ARC)
using the points (6, 44) and (20, 37). Analyze the units of the ARC to
inter pret your results .

5.2 Graphs of Linear Equations (Straight Lines)

5.2.1 Linear Equations in Two Variables in the form
y =mx +b

A linear equation in two variables is any equation whose graph
forms a straight line. One form of a linear equation in two variables is
y = mx + b, where x and y are the variables, m is the [constant]
coefficient of x, and b is also a constant. A second form will be
discussed in the next section.

Example 1 For each equation, identify the constants m and b.

Equation, y = mx + b m b

Solution to a Linear Equation in Two Variables
A solution to a linear equation in two variables is an ordered pair
of numbers (x, y) that [when substituted for the variables] makes the
equation true.

Example 2 Is (3, -5) a solution to y = -3x + 4?

Example 3 Is a solution to y = -3x + 4?

5.2.2 Graphs of Linear Equations in the form y =mx +b
Equations with two variables have an infinite number of ordered-pair
solutions. The graph of an equation in two variables is a drawing
of all the ordered-pair solutions to the equation. The graph of every
linear equation in two variables (i.e.y = mx + b) forms a straight line.

Graphing Linear Equations by Plotting Three Points
To graph a linear equation in the form y = mx + b:
1. Pick three values for x .
2. For each x chosen in step 1, use the given equation, y = mx + b, to
compute the three corresponding values of y.
3. Plot the three ordered pairs and use a ruler to draw a straight line
through the points. Draw the line through the entire grid and draw
arrowheads at each end of the line to show that the line continues

Example 5
Graph y = 2x - 3 by
finding three solutions.
Example 6
Graph y = -3x + 2 by
finding three solutions.
Example 7
 Graph by
finding three solutions.
Example 8
Graph by
finding three solutions.


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