Slopes and Equations of Lines
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Let's review our knowledge of slopes
and equations of lines. 

Slopes Equations of lines

The slope of a line is a rate of change and is represented by m. 
        

 When a line passes through the points
 (x1, y1) and (x2, y2),  the slope is
m = .
Lines that have a positive slope, rise from lower left to upper right.  They go up hill.
Lines that have a negative slope, decline from upper left to lower right.  They go down hill.
Lines that are horizontal have a slope of zero.  (There is no "rise", creating a zero numerator.)
Lines that are vertical have no slope (undefined slope).  (There is no "run", creating a zero denominator.)
Lines that are parallel have equal slopes.
Lines that are perpendicular have negative reciprocal slopes.
(such as m = 2 with m = -1/2)


Equations of line can take on several forms:


Slope Intercept Form:
[used when you know, or can find, the slope, m, and the y-intercept, b.]
           y = mx + b
 

Point Slope Form:
[used when you know, or can find, a point on the line (x1, y1), and the slope, m.]
               


Horizontal Line Form:
           y
= 3
(or any number)
Lines that are horizontal have a slope of zero.  They have "run", but no "rise".   The rise/run formula for slope always yields zero since rise = 0.
                                  y = mx + b
                                  y = 0x + 3
                                  y = 3
 

Vertical Line Form:
           x = -2
(or any number)
Lines that are vertical have no slope (it does not exist).  They have "rise", but no "run".  The rise/run formula for slope always has a zero denominator and is undefined.

 

 

Examples:

1.  Find the slope and y-intercept for the equation 3y = -9x + 15.

First solve for "y =":      y = -3x + 5
Use the form:     y = mx + b
Answer:  the slope (m) is -3
                the y-intercept (b) is 5
 

2.  Find the equation of the line whose slope is 4 and crosses the y-axis at (0,2).

In this problem m = 4 and b = 2.
Use the form:  y = mx + b
Substitute:           y = 4x + 2    

3.  Given that the slope of a line is -3 and the line passes through the point (-2,4), write the equation of the line. 

The slope:  m = -3
The point (x1 ,y1) = (-2,4)
Use the form:  y - y1 = m ( x - x1)
                     y - 4 = -3 (x - (-2))
                     y - 4 = -3 ( x + 2)   ANS.
If asked to express the answer in "y =" form:
                          y - 4 = -3x - 6
                          y = -3x - 2

4.  Find the slope of the line that passes through the points (-3,5) and (-5,-8).

First find the slope:   

Using either point:  (-3,5)
Remember the form:  y - y1 = m ( x - x1)
Substitute:  y - 5 = 6.5 ( x - (-3))
                  y - 5 = 6.5 (x + 3)  Ans.
 

5.  Given that the line is parallel to y = 4x + 5 and passes through the point (-2,4), write the equation of the line. 

Parallel lines have equal slopes, so m = 4.
The point (x1 ,y1) = (-2,4)
Use the form:  y - y1 = m ( x - x1)
                     y - 4 = 4(x - (-2))
                     y - 4 = 4 ( x + 2)   ANS.
 

4.  Given 2y = 6x + 12 and 3y + x = 15, determine if the lines are parallel, perpendicular, or neither.

Put in "y=" form to observe the slopes.
2y = 6x + 12 gives y = 3x + 6, so m = 3
3y + x = 15 gives y = -1/3 x + 5, so m = -1/3
Since the slopes are negative reciprocals, the lines are perpendicular. ANS.