Equations of Straight Lines
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When working with straight lines, there are several ways to arrive at an equation which represents the line.   


Remember:

Slope is found by using the formula:
 


Slope is also expressed as rise/run.



Equation Forms of Straight Lines

Slope Intercept Form

Point Slope Form

Use this form when you know the slope and the y-intercept (where the line crosses the y-axis).

y = mx + b

m = slope
b
= y-intercept
 (where line crosses the y-axis.)

Use this form when you know a point on the line and the slope (or can determine the slope).

m = slope

= any point on the line

 

Horizontal Lines

Vertical Lines

y = 3 (or any number)
Lines that are horizontal have a slope of zero.  Horizontal lines have "run", but no "rise".   The rise/run formula for slope always yields zero since the rise = 0.
Since the slope is zero, we have
y = mx + b
y = 0x + 3
y = 3
This equation also describes what is happening to the y-coordinates on the line.  In this case the y-coordinates are always 3.

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

The equations for these lines describe what is happening to the x-coordinates.  In this example, the x-coordinates are always equal to -2.


Examples:

Examples using Slope-Intercept Form:

Examples using Point-Slope Form:

1.  Find the slope and y-intercept for the equation 2y = -6x + 8.

First solve for "y =":      y = -3x + 4
Remember the form:     y = mx + b
Answer:  the slope (m) is -3
                the y-intercept (b) is 4

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)
Remember the form:  y - y1 = m ( x - x1)
Substitute:                 y - 4 = -3 (x - (-2))
ANS.                        y - 4 = -3 ( x + 2)  

If asked to express the answer in "y =" form:             y - 4 = -3x - 6
                          y = -3x - 2

2.  Find the equation of the line whose slope is 4 and the coordinates of the y-intercept are (0,2).

In this problem m = 4 and b = 2.
Remember the form:  y = mx + b and that b is where the line crosses the y-axis.
Substitute:           y = 4x + 2    

 

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

First, find the slope:   

Use 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.