Overview


In this lesson, you will review how
to graph linear equations on a
Cartesian coordinate system . You will also find the x and yintercepts,
the
slope of a line, and the distance between two points.
Finally, you will review how to write the equation of a line in three
forms:
pointslope form, standard form, and slope intercept form . 
Explain

Concept 1 has sections on
• The Cartesian Coordinate
System
• The Distance Formula
• Graphing a Linear Equation
• Finding x and yintercepts
• Horizontal and Vertical
Lines
• The Slope of a Line

CONCEPT 1:
GRAPHING LINESThe Cartesian
Coordinate System
The Cartesian coordinate system consists of two real number lines
placed at right angles to each other.
The horizontal number line is called the xaxis.
The vertical number line is called the yaxis.
The axes define a flat surface called the xyplane.
Every point in the xyplane has two numbers associated with it.
• The xcoordinate or abscissa tells how far the point lies to the left
or
right of the yaxis.
• The ycoordinate or ordinate tells how far the point lies above or
below
the xaxis.
The xcoordinate and the ycoordinate are often written inside
parentheses ,
like this: (x, y).
The first number, x, represents the xcoordinate and the second number,
y, represents the ycoordinate.
For example, the point that is 3 units to the
right of the yaxis and 6 units
below the xaxis is labeled (3, 6).
Because the order in which the pair of numbers is written is important,
(x, y) is called an ordered pair. Thus, the point ( 6, 3) is not the
same as
the point (3, 6).
The xaxis and the yaxis intersect at the point (0, 0). This point is
called
the origin.
The xaxis and the yaxis divide the xy plane into four regions called
quadrants.
Quadrant 
Sign of x 
Sign of y 
I 
positive 
positive 
II 
negative 
positive 
III 
negative 
negative 
IV 
positive 
negative 
A point on an axis does not lie in a quadrant. 

Example EII.E.1
Find the coordinates of each point
labeled on the graph.
Then, state the quadrant in which
each point lies.
Solution
Point A has coordinates (2, 5), and lies in Quadrant I.
Point B has coordinates (3, 4), and lies in Quadrant IV.
Point C has coordinates ( 5, 3), and lies in Quadrant II.
Point D has coordinates ( 6, 2), and lies in Quadrant III.
Point E has coordinates (5, 0). It is not in a quadrant since it lies on
the
xaxis. 
Example EII.E.2
Plot each point on a Cartesian coordinate system:
Solution
The plot the point (5, 4), start at the origin:
• move 5 units to the right;
• then move down 4 units;
• place a dot at this location.
Follow a similar procedure for the other points. Notice the difference
between the locations of points ( 3, 0) and (0, 3). 

The Distance Formula
The distance between any two points in the xyplane can be found using
the distance formula.
— Formula —
The Distance FormulaLet (x_{1},
y_{1}) and (x_{2}, y_{2}) represent any
two points in the xyplane. The
distance, d, between the points is given by
If the points lie on a horizontal line,
then this simplifies to
If the points lie on a vertical line, then
this simplifies to


In the distance formula, it doesn’t
matter
which point is considered (x_{1}, y_{1}) or (x_{2},
y_{2}).
The resulting distance is the same. 
Example EII.E.3
Find the distance between ( 3, 4) and (5, 4).
Solution
Since the points ( 3, 4) and (5, 4) have the same
ycoordinate, 4, they lie on a horizontal line. So,
we use the formula: 

Let x_{1} =3 and x_{2}
=5. 

Simplify. 

Find the absolute value. 

The distance between ( 3, 4) and (5, 4) is 8
units 
In the formula, ,
it doesn’t
matter which xcoordinate is assigned to
x_{1} and which is assigned to x_{2}. The
resulting distance is the same.
That is:

If we switch the points and let
and
, we get the
same answer.We can use a calculator to
approximate 
Example EII.E.4
Find the distance between ( 3, 8) and (5, 1).
Solution
The points do not lie on a horizontal or vertical line
Therefore, use the distance formula. 

Let
and
. 

Substitute. 

Simplify. 

The distance between the points ( 3, 8) and (5,
1) is units. 
Since two points determine a line,
you can
use any two points that satisfy the equation
to draw the line. Plotting more than two
points will help avoid errors. 
Graphing a Linear Equation
An equation that can be written in the form
is called a
linear
equation because its graph is a straight line.
Here are some examples:



(Here, A 0.) 
(Here, B 0.) 
We can graph a linear equation on a Cartesian
coordinate system by
plotting points that satisfy the equation.
— Procedure —
To Graph a Linear Equation
Step 1 Make a table of ordered pairs that satisfy the
equation.
Step 2 Plot the ordered pairs.
Step 3 Draw a line through the plotted points. 

Here’s how to calculate the
corresponding
value for y when x = 3.
Equation.
Replace x with 3.
Multiply.
Add 12 to both sides.
Divide both sides by 6.

Example EII.E.5
Graph the linear equation
.
Solution
Step 1 Make a table of ordered pairs that satisfy the equation.
Select values for x and then use
to calculate the
corresponding values for y.
Step 2 Plot the ordered pairs.
The points are shown on the graph.
Step 3 Draw a line through the plotted points.
Each point on the line represents a solution of the equation

Finding x and yintercepts
The point where a graph crosses an axis called an intercept.
— Definition —
xintercept and yintercept 
The xintercept is the point
where the line crosses the xaxis.
The xintercept has the form
(a, 0), where a is a constant.
The yintercept is the point
where the line crosses the yaxis.
The yintercept has the form
(0, b), where b is a constant. 


The xintercept lies on the xaxis,
so y= 0.
The yintercept lies on the yaxis, so x= 0. 
Example EII.E.6
Given the equation
a. Find the xintercept.
b. Find the yintercept.
c. Use the intercepts to graph the line.
Solution
a. The xintercept has the form (a, 0). 

To find the xintercept, substitute 0 for
y. 

Then, solve for x . 

The xintercept is (4, 0). 


b. The yintercept has the form (0, b). 

To find the yintercept, substitute 0 for
x. 

Then, solve for y. 

The yintercept is (0, 2). 


c. To graph the line
,
first plot
the x and yintercepts.

x 
y 

Then, draw a line through the
intercepts. 
4 
0 
←xintercept 
As a check, it is a good idea to
find a third 
0 
2 
←yintercept 
point on the line. 
2 
3 
←check point 
For example, let x = 2 in .
Then, solve for y.
The result is y = 3. So, the point ( 2, 3) should also lie on
the line. 

Two points that are
often easy to find are
the x and yintercepts. 


Horizontal and Vertical Lines
The graph of a linear equation,
, where A and
B are
not both 0, is a straight line.
• If A =0, then the graph is a horizontal line. For example, y= 3 is a
horizontal line. On this line, the ycoordinate is always 3.
• If B =0, then the graph is a vertical line. For example, x =5 is a
vertical line. On this line, the xcoordinate is always 5 

Example EII.E.7
Graph the following linear equations:
Solution
a. The graph of y 4 is a horizontal line.
On this line, the ycoordinate is always 4.
The yintercept is (0, 4); the line has no xintercept. 

b. The graph of x =4 is a vertical line.
On this line, the xcoordinate is always 4.
The xintercept is (4, 0); the line has no yintercept. 



The Slope of a Line
The rise of a line represents the vertical change when moving from one
point to a second point on a line.
The run of a line represents the horizontal change when moving from one
point to a second point on the line.
The slope of a line is the ratio of the rise to the run.
It is a number that
describes the steepness of the line.
— Definition —
Slope of a Line
The slope of the line that passes through two points, (x_{1},
y_{1}) and
(x_{2}, y_{2}), is given by
slope
where 

Example EII.E.8
Find the slope of the line that passes through the
points (2, 7) and (4, 3).
Solution
Let
and
. 

Substitute these values in the slope
formula. 

Simplify. 

Reduce . 

Thus, the slope of the line through ( 2, 7) and
(4, 3) is . 
When using the slope formula, it does not
matter which point we choose for (x_{1}, y_{1})
and which we choose for(x_{2}, y_{2}). 
Example EII.E.9
Find the slope of the line that passes through the
points ( 4, 5) and (3, 5).
Solution
Let
and
. 

Substitute these values in the slope
formula. 

Simplify. 

Divide. 

Thus, the slope of the line through ( 4, 5) and
(3, 5) is 0.
In fact, the slope of any horizontal line is 0. 

Example EII.E.10
Find the slope of the line that passes through the
points (3, 2) and (3, 4).
Solution
Let
and
. 

Substitute these values in the slope
formula. 

Simplify. 

Since division by zero is undefined , the slope is
undefined.
In fact, the slope of any vertical line is undefined. 


— Summary —
Slope of a Lin 
Slope is positive: Line slants
upward as we move from left to
right.
• Slope is negative : Line slants
downward as we move from left
to right.
• Slope is zero: The slope of a
horizontal line is 0.
• Slope is undefined: The slope
of a vertical line is undefined. 

We can use slope to help us construct the graph of
a line. 

Example EII.E.11
Graph the line that passes through the point ( 5, 6)
with slope
.
Solution
First, plot the given point, ( 5, 6).
To find another point on the line, use the slope. The slope,
,
tells us how to move up and down (rise) and left and right (run) to get
to
another point on the line. The slope says
to move 3 units
down and 4 units right to get to another point, ( 1, 3).
Plot the point ( 1, 3).
Finally, draw a line through the two points. 

Here is a summary of this concept
from Interactive Mathematics.
