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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 Other problems will be posted by request! Dr. Robert J. Rapalje Seminole Community College Sanford, FL 32773
18.
Find
the equation of the line through 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 Start with
the formula:
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:
Final answer :
20. Find
the equation of the line through 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 Now find
the equation of a line with Start with
the formula:
Be sure to
answer the question! Find the equation of the line
Check your
answer by substituting
Final answer:
21.
Find the equation of the
line through 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 Now find
the equation of a line with Start with
the formula:
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
Final
answer:
22. Find the equation of the line through 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 Now find
the equation of a line with Start with
the formula:
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
Final answer :
24.
Find the equation of the line through 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
Divide both sides by -4:
The slope
of the given line is Now find
the equation of a line with Start with
the formula:
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
Final answer :
27.
Find the equation of the
perpendicular bisector
of the line segment between the two given points:
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, 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
Start
with the formula:
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
Final
answer:
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