December 8th
December 8th
Graph of a Line
We can graph the linear equation defined by y = x + 1 by
finding several ordered pairs. For example, if x = 2 then y = 2 +
1 = 3, giving the ordered pair (2, 3). Also, (0, 1), (4, 5), (2,
1), (5, 4), (3, 2), among many others, are ordered pairs
that satisfy the equation.
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To graph y = x + 1 we begin by locating the ordered pairs
obtained above, as shown in Figure 6(a). All the points of this
graph appear to lie on a straight line, as in Figure 6(b). This
straight line is the graph of y = x + 1.
It can be shown that every equation of the form ax + by = c
has a straight line as its graph. Although just two points are
needed to determine a line, it is a good idea to plot a third
point as a check. It is often convenient to use the x and
yintercepts as the two points, as in the following example.
Example
Graph of a Line
Graph 3x + 2y = 12.
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Solution
To find the y intercept, let x = 0.
3(0) + 2y = 12 

2y = 12 
Divide both sides by 2. 
y = 6 

Similarly, find the xintercept by letting y = 0 which gives x
= 4. Verify that when x = 2 the result is y = 3. These three
points are plotted in Figure 7(a). A line is drawn through them
in Figure 7(b).
Not every line has two distinct intercepts; the graph in the
next example does not cross the xaxis, and so it has no
xintercept.
Example
Graph of a Horizontal Line
Graph y = 3.
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Solution
The equation y = 3 or equivalently y = 0x 3, always gives
the same y value,  3, for any value of x . Therefore, no value
of x will make y = 0, so the graph has no x intercept. The graph
of such an equation is a horizontal line parallel to the x axis.
In this case the y intercept is  3, as shown in Figure 8.
In general, the graph of y = k, where k is a real number, is
the horizontal line having yintercept k.
The graph in Example 13 had only one intercept. Another type
of linear equation with coinciding intercepts is graphed in
Example 14.
Example
Graph of a Line Through the Origin
Graph y = 3x.
Solution
Begin by looking for the x intercept. If y = 0 then
y = 3x 

0 = 3x 
Let y = 0 
0 = x 
Divide both sides by 3 
We have the ordered pair (0, 0). Starting with x = 0 gives
exactly the same ordered pair, (0, 0). Two points are needed to
determine a straight line, and the intercepts have led to only
one point. To get a second point, choose some other value of x
(or y ). For example, if x = 2 then
y = 3x = 3(2) = 6, (let x = 2)
giving the ordered pair (2, 6). These two ordered pairs, (0,
0) and (2, 6), were used to get the graph shown in Figure 9.
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