The graphs of the equations shown at the right are a family of graphs because they have the same slope. Because 4x is never equal to 4x + 5, the value of y will never be the same for any given value of x, and the graphs will never intersect. These lines are parallel.
1.
Determine whether the graphs of the equations are parallel.
Alternative Solutions:
First, determine the
slopes of the lines. Write each equation in slope-intercept form.
The slopes are the same,
so the lines are parallel. Check by graphing.
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2.
Determine whether figure ABCD is a parallelogram.
Alternative Solutions:
You can use
the slope of a line to write an equation of a line that is parallel to it.
Example
3.
Write an equation in slope-intercept form of the line that is parallel to
the graph of y = –4x + 8 and passes
through (1, 3).
Alternative Solutions:
The slope of the given
line is –4. So, the slope of the new line will also
be –4. Find the new equation by using the point-slope form.
An equation whose graph
is parallel to the graph of 4x + y = 8 and passes through (1, 3)
is y = –4x + 7. Check by
substituting (1, 3) into y = –4x + 7 or by graphing.
The results
of the Hands-On Algebra activity lead to the following definition of perpendicular lines.
Example
4.
Determine whether the graphs of the equations are perpendicular.
Alternative Solutions:
The graphs are
perpendicular because the product of their slopes is 
.
Example
5.
Write an equation in slope-intercept form of the line that is perpendicular
to the graph of 
and passes through (–4, 2).
Alternative Solutions:
The slope is ⅓. A line
perpendicular to the graph of has slope –3. Find the new equation by using the point-slope
form.
The new equation is y
= –3x – 10. Check by substituting (–4, 2) into the
equation or by
graphing.
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