Slopes and Equations of Lines Topic Index | Geometry Index | Regents Exam Prep Center

 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.

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