5.03  Equation of a Line

Intermediate Algebra: One Step at a Time,  Page 403:  24, #33a) and b), 34a) and b), Extra Problem

Other problems will be posted by request!

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

 

 

p. 401.  # 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:      

 

Page 403, # 33a)   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: 

 

p. 403.  # 33b)   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: 

 

 

p. 404.  # 34a)  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:                                 

p. 403.  # 34b)  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 

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.  Your first step, obviously, is to find the slope of the line.  You must use the formula for the slope between two points:

                        

Find the slope of the given line by solving for y in terms of x.   

                       

                   

                   

                   

Now that you have found the slope, there are two methods of finding the equation of the line.  While I used to solve these problems by the method of the “point-slope formula,” I believe another method is easier.  Since this is MY website, I’ll show you MY favorite method first, then I’ll use the other method as well.  Let me know what YOU think!!

 

The method:

The equation  has four unknowns:  x and y, which are variables, and  and  which are constants.  You know that , and you need to find .  You can use either given point, either  or   for the values of x and y, and  to find the value of .

                      , where , , and  

 

                     

You must solve for :

                     

So,                      and the equation is

                       

                      

You can check this answer by substituting the OTHER values of x and y, that is  into the equation to see if it actually works:

                      

                       

                      

IT CHECKS!!

 

Point-Slope Formula Method

This method uses the famous point-slope formula:

                      ,

where   represents any point on the line and having already found that .  Let , .

                    

             

                    

                    

 

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Dr. Robert J. Rapalje Altamonte Springs Campus
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