Analyzing Equations of a Line

Linear Equations which describe linear functions come in three (3) basic formats.

  1. "Slope-intercept" form: y = mx + b
  2. "Point-slope" form : y2 - y1 = m(x2 - x1)
  3. "Standard" form: ax + by = c
All forms can be used to describe a specific line to agree with specific requirements. All ways should be practiced.

Example (1):Write the equation of a line through (3, 5) with slope (1/2)

Solution: The most convenient form given the information provided is "point-slope": substitute into the basic form to get..... y - 5 = (1/2)(x - 3)


Example (2):Write the equation of the line through (5, 1) and (-2, 7)

Solution:Use "point-slope" form after calculating the slope using the slope formula.
Slope = (7 - 1)/(-2 - 5) = 6/-7 = -(6/7)
Select either point and substitute into point-slope form:
          y - 1 = (-6/7)(x - 5)         or....
          y - 7 = (-6/7)(x + 2)
Note: These equations can easily be transformed into "slope-intercept" form by distributing and writing in y= form.

Example (3):Write 3x + 2y = 8 in "slope-intercept" form.

Solution: 
           3 x + 2y = 8
                 2y = 8 - 3x
                 2y = -3x + 8
                  y = (-3x + 8)/ 2      or.....
                  y = (-3/2)x + 4


Should the information come to you in table or graph form you must...
  • Get your own data points
  • Follow the procedure in one of the examples above



Parallel lines...Perpendicular lines and their slopes

Information concerning the slope of a line may come in a "back door" manner,
ie. your line is parallel to the line y = 4x -1 or your line is perpendicular to the line y = 4x - 1
It is expected that you understand that lines become parallel only when they have equal (same) slopes. Lines which are perpendicular have related slopes also; the numbers are negative reciprocals of each other, ie. (-3) and (+1/3)


Example (4):Write the equation of the line parallel to y = 4x - 1 through (5, -2). Then find the equation of the line perpendicular to y = 4x - 1 through (5, -2)

Solution: 
            Since the desired line is parallel to y = 4x - 1, its slope is also 4
            The line must also contain the point (5, -2) as requested
            We can use "point-slope" form now to get OUR line:  y + 2 = 4(x - 5)


            The second request is for a line perpendicular to the given line
            The slope of this perpendicular line is (-1/4) by the above explanation
            We use "point-slope" form to get OUR line:  y + 2 = (-1/4)(x - 5)


            This can be converted to "slope-intercept" form as follows:
                      y + 2 = (-1/4)(x - 5)
                      y + 2 = (-1/4)x + (5/4)
                      y     = (-1/4)x -(3/4)


                      
            We can continue and convert to "standard" form which prohibits fractional
            coefficients and requires all varaibles to be one the side of =.

                     4y     =   -x - 3
               x  +  4y     =   -3




[TOC] [Mathematical "Models"] [Functions] [Probability] [Direct Variation] [Solving Linear Equations]
[Analyzing Equations of Lines] [One Variable Inequalities] [Arithmetic Sequences] [Geometric Sequences] [Irrational Numbers] [Complex Numbers] [Quadratic Functions] [Solving Quadratic Equations] [Conic Sections] [Variation] [Exponents and Roots] [Solving Radical Equations] [Function Operations] [Polynomial Functions] [Rational Expressions] [Rational Functions] [Solving Rational Equations] [Exponential Functions] [Logarithmic Functions]