2.01 Linear
Graphs, Slope, and the Equation
of a Line
College Algebra: One Step at a Time,
Page
209-211: #18, 20, 21, 22, 24, 27
Dr. Robert J. Rapalje
Seminole State College of Florida
Altamonte Springs Campus
To see
Section 2.01, with
detailed explanations, examples, exercises, and answers,
click here!
18.
Find
the equation of the line through 
and
Solution:
In order
to find the equation of any line, you must have a point (from which to
start!) and a slope (a direction in which to go!). In this case, you are
given two points. The first step is to find the slope between these
two points.
Remember
the formula for the slope between two points
The slope
of the given line is 
Now find
the equation of a line with passing
through either of the given points. It doesn’t matter which point you use.
Let’s use the first point 
.
Start with
the formula: 
, where
,
.
To clear
the fraction, multiply by the denominator which is
.
Divide out
the :
Subtract 
:
Divide by
:
Be sure to
answer the question! Find the equation of the line
Check your
answer by substituting the values of the other point: 
to
see if 
It checks!!
Final answer :
20. Find
the equation of the line through 
and
parallel to
.
Solution:
In order
to find the equation of any line, you must have a point (from which to
start!) and a slope (a direction in which to go!). In this case, you are
given a point, but instead of being given the slope of the line, you are
given the equation of a given line that is
parallel
to it. Since the
lines are parallel, they have the
same slope!!
The slope
of the given line is 
,
so the slope of any line
parallel
to this line is also
.
Now find
the equation of a line with passing
through.
Start with
the formula: , where
,
.
Be sure to
answer the question! Find the equation of the line
Check your
answer by substituting 
to
see if 
It checks!!
Final answer:
21.
Find the equation of the
line through 
and
parallel
to
.
Solution:
In order
to find the equation of any line, you must have a point (from which to
start!) and a slope (a direction in which to go!). In this case, you are
given a point, but instead of being given the slope of the line, you are
given the equation of a given line that is
parallel
to it. Since the lines are
parallel,
they have the same
slope!!
The slope
of the given line is 
,
so the slope of any line
parallel
to this line is also
.
Now find
the equation of a line with passing
through .
Start with
the formula: , where
,
.
In this
problem, you might want to multiply both sides of the equation by the
denominator which is 4, and if you do it will be correct! However, notice
that the 4 in the denominator divides out with the other 4 in the product,
and the result is just 3. Isn’t this easier?
Be sure to
answer the question! Find the equation of the line
Check your
answer by substituting 
to
see if 
It checks!!
Final
answer:
22. Find the equation of the line through and
perpendicular to.
Solution:
In order
to find the equation of any line, you must have a point (from which to
start!) and a slope (a direction in which to go!). In this case, you are
given a point, but instead of being given the slope of the line, you are
given the equation of a given line that is
perpendicular
to it.
Since the lines are perpendicular, one slope must be the
negative
reciprocal of the other.
The slope
of the given line is ,
so the slope of a line
perpendicular to this line is must be
.
Now find
the equation of a line with passing
through.
Start with
the formula: , where
,
.
To clear
the fraction, multiply by the denominator which is
.
Divide out
the :
Add 
:
Divide by
:
Be sure to
answer the question! Find the equation of the line
Check your
answer by substituting to
see if
It checks!!
Final answer :
24.
Find the equation of the line through and
perpendicular
to
.
Solution:
In order
to find the equation of any line, you must have a point (from which to
start!) and a slope (a direction in which to go!). In this case, you are
given a point, but instead of being given the slope of the line, you are
given the equation of a given line. Your line must be
perpendicular
to this given line, which means that the given line has a slope which is the
negative reciprocal
of the slope of the line you need to find.
First you
must find the slope of the given line by solving for y in
terms of x.
Add
to each side of the equation:
Divide
both sides by -4:
The slope
of the given line is 
, so the slope of a line
perpendicular to this
line is also 
.
Now find
the equation of a line with passing through.
Start with
the formula: ,
where , .
To clear
the fraction, multiply by the denominator which is
.
Divide out
the : 
Subtract
:
Divide by
:
Be sure to
answer the question! Find the equation of the line
Check your
answer by substituting to see if
It checks!!
Final answer :
27.
Find the equation of the
perpendicular bisector
of the line segment between the two given points:
and
.
Solution:
It isn’t
required, but it might help to draw a sketch of these two points, and draw
the line segment between them. You can start by finding the slope of this
line segment. Remember the formula for the slope between two points
The slope
of this line segment is 
Now, you
need to find the
perpendicular bisector
of this line segment. The
perpendicular bisector will be a line that passes through the
midpoint of these two points,
and it will be
perpendicular
to the line segment. Do
you remember how to find the midpoint between two points? It’s like the
average of the x coordinates and the average of the y coordinates, so you
add the x coordinates together and divide by 2, and add the y coordinates
and divide by 2:
Next, you
know that the perpendicular bisector of this line segment will be
perpendicular to the line segment. Since the slope of the line segment is
already known, ,
the slope
of the perpendicular
bisector will be the
negative reciprocal
of the slope
of this line segment, which will be
.
Here is a
place to be careful. In previous problems in which you were trying to find
the equation of a line between two points, it didn’t matter which point you
used in the formula. In this case there are three points, and at first
glance you might think that, like before, one point will work as well as any
other point. However, look at the sketch of the two given points and the
midpoint. All three points lie on the line segment, but how many of these
points actually lie on the perpendicular bisector of the segment? Answer:
only the midpoint!
Therefore, you MUST use the midpoint. Do NOT use the end points of the line
segment because these points are NOT on the perpendicular bisector!!
So you
must now find the equation of a line with
passing
through
.
Start
with the formula: , where
,
.
Multiply
both sides of the equation by 5:
Be sure
to answer the question! Find the equation of the line
Check
your answer by substituting 
to
see if 
It checks!!
Final
answer:
Return to main page
Math in Living
C
O
L O
R
!!
Return to Intermediate Algebra page
Return to
College Algebra page
