A simple method of graphing a linear equation is using the points where the line crosses the x-axis and the y-axis. Consider the equations x + y = 8.
The x-intercept
is 8, and the y-intercept is 8. This means that the graph intersects the
x-axis at (8, 0) and the y-axis at (0, 8). Graph these ordered
pairs. Then draw the line that passes through these points. Each point on the
graph represents two numbers whose sum is 8. Notice that the table of values confirms
these intercepts.
Example
Determine
the x-intercept and y-intercept of the graph of each equation.
Then graph the equation.
1.
5y – x = 10
Alternative Solutions:
The x-intercept is
–10, and the y-intercept is 2. This means
that the graph intersects the x-axis at (–10, 0) and the y-axis
at (0, 2). Graph these ordered pairs. Then draw the line that passes through
these points.
.PNG)
2.
2x – 4y = 8
Alternative Solutions:
The x-intercept is
4, and the y-intercept is –2. This means that the graph intersects the x-axis
at (4, 0) and the y-axis at (0, –2). Graph these ordered pairs. Then
draw the line that passes through these points.
.PNG)
Check: Look at the graph. Choose some other point on the line
and determine whether it is a solution of 2x – 4y = 8. Try (2, –1).
You can easily find the
slope and y-intercept of the graph of an equation that is written in
slope-intercept form.
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3.
To mail a letter in 2004, it cost $0.37 for the first ounce and $0.23 for
each additional ounce. This can be represented by y = 0.37 + 0.23x.
Determine the slope and y-intercept of the graph of the equation.
Alternative Solutions:
The slope is 0.23, and the y-intercept
is 0.37. So the slope represents the cost per ounce after the first ounce, and
the y-intercept represents the cost of the first ounce of mail.
Example
4.
Determine the slope and y-intercept of the graph of 10 + 5y
= 2x.
Alternative Solutions:
Write the equation in
slope-intercept form to find the slope and y-intercept.
The slope is 
, and the y-intercept is –2.
You can also
graph a linear equation by using the slope and y-intercept.
Example
Graph
each equation by using the slope and y-intercept.
Alternative Solutions:
The slope is 
, and the y-intercept is –5.
Graph the point at (0, –5). Then go up 2 units and right 3 units. This will be
the point at (3, –3). Then draw the line through points at (0, –5) and (3, –3).
Check: The graph appears to go through the point at (6, –1). Substitute
(6, –1) into 
.
6.
3x
+ 2y = 6
Alternative Solutions:
First, write the equation
in slope-intercept form.
Graph the point at (0, 3). Then go up
3 units and left 2 units. This will be the point at (–2, 6). Then draw the line
through (0, 3) and (–2, 6). Check by
substituting the coordinates of another point that appears to lie on the line,
such as (2, 0).
The graph of a horizontal line has a slope of 0 and no x-intercept.
The graph of a vertical line has an undefined slope and no y-intercept.
Example
Graph each equation.
Alternative Solutions:
.PNG)
No matter what the value of x, y = 4. So, all ordered pairs are of
the form (x, 4). Some examples are (0, 4) and (–3, 4).
8. x =
–2
Alternative Solutions:
slope: undefined, y-intercept: none
No matter what the value of y, x = –2. So, all ordered
pairs are of the form (–2, y). Some examples are (–2, 1)
and (–2, 3).
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