4.06  Equation of a Line

Basic Algebra: One Step at a Time,  Page 341 - 348:  24, #31a) and b), Extra Problem

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

24.   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 the two points, using the well-known formula for the slope between two points:

Now, write down the formula : , where .  You can use either point, let’s say .

To clear the fraction, multiply by the denominator which is .

Divide out the 7:     

Solve for b:              

Be sure to answer the question!  Find the equation of the line 

Check your answer, be substituting to see if .

                                                              It checks!!

Final answer:      

31a)   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.  Your line must be parallel to this given line, which means that the given line has the same slope as the line you need to find.

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 3:

The slope of the given line is   , so the slope of a line parallel 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 3:          

Solve for b:                  

Be sure to answer the question!  Find the equation of the line 

Check your answer, be substituting to see if

                                                              It checks!!

Final answer: 

31b)   Find the equation of the line through    and perpendicular to.

Solution:     In the previous part of this problem, you found the slope of the given line to be , so the slope of a line perpendicular  to this line is the negative reciprocal of this slope, which is .

Now find the equation of a line with  passing through,

Start with the formula: , where , .

In this case, the fraction clears itself, so you can just divide out the 4.  However, if you prefer to work it like most of the other problems like this, you can clear the fraction as before.  The result will be the same.  HOWEVER, let’s do it the easy way!!

Divide out the 4 :         

Solve for b:                  

Be sure to answer the question!  Find the equation of the line 

Check your answer, be substituting to see if

Divide out the 4:

                                                              It checks!!

Final answer: 

Extra Problem:    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.  Your line must be parallel to this given line, which means that the given line has the same slope as the line you need to find.

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 :

The slope of the given line is   , so the slope of a line parallel 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 3:     

Solve for b:              

Be sure to answer the question!  Find the equation of the line 

Check your answer, be substituting to see if

                                                              It checks!!

Final answer:                                 

b)  Find the equation of the line through    and perpendicular to.

Solution:    

In the previous part of this problem, you found the slope of the given line to be , so the slope of a line perpendicular  to this line is the negative reciprocal of this slope, which is .

Now find the equation of a line with  passing through,

Start with the formula: , where , .

In this case, the fraction clears itself, so you can just divide out the 4.  However, if you prefer to work it like most of the other problems like this, you can clear the fraction as before.  The result will be the same.  HOWEVER, let’s do it the easy way!!

Divide out the 4 :    

Solve for b:              

Be sure to answer the question!   Find the equation of the line 

Check your answer, be substituting to see if

Divide out the 4:

                                                                   It checks!!

Final answer:                                       .

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