Linear Equations, Part 1: Basics

With finals coming up I have been doing a lot of linear equation problems with my students, so I thought it appropriate to do a post about linear equations. One notable characteristic about linear equations is that they have constant slopes. This leads to a few simple equations and techniques that will allow the student to solve any linear problem.

First, let's talk about the three forms of linear equations:

*Standard Form: Ax + By = C
*Slope-Intercept Form: y = mx + b, where b is the y-intercept
*Point-Slope Form: y - y1 = m(x - x1), where (x1,y1) is a known coordinate.

In these equations m represents the slope. If there is no information given about the line except for two points (x1,y1) and (x2,y2), then the slope can be found using the slope formula:

  m = (y2-y1)/(x2-x1), also known as "rise over run"

We can then put the slope and one of the known points into the Point-Slope form. Solving for y will yield the Slope-Intercept form. If necessary we could rearrange the Slope-Intercept form into the Standard form.

Example:


Given the two points (2,6) and (6,8) find the equation of the line that passes through these points,
a) in slope-intercept form
b) in standard form
c) state the coordinate of the y-intercept

Solution

a) First we have to find the slope. Using the slope formula we have:
           m = (8-6)/(6-2) = 2/4 = 1/2  *always simplify fractions when possible
   
    Now that we know the slope, we substitute it and one of the points into the point-slope form:
          y - 6 = (1/2)(x - 2)  *either known point will work. I chose (2,6) because of the smaller numbers
    
     Solve for y
          y - 6 = (1/2)x - 1  *distributive property
          y = (1/2)x + 5       *add 6 to both sides to isolate y
     
     The slope-intercept form is y = (1/2)x + 5

b) All we have to do here is rearrange the slope-intercept form, solving for the constant
         y = (1/2)x + 5   *start with slope-intercept form
         -(1/2)x + y = 5  *subtract (1/2)x from both sides to isolate the constant.
         2[-(1/2)x + y] = 5(2)   *multiply both sides by 2 to clear the fraction
         -x + 2y = 10   *Standard form

c) Refer to the slope-intercept form, y = (1/2)x + 5
          In this instance b = 5, so the coordinate of the y-intercept is (0,5)  *x is always zero at the y-intercept

So we have gone from knowing nothing but two points on a particular line and was able to determine the slope, all three forms of the equation, and identify the y-intercept. All of this is possible because linear equations have constant slopes.

In Part 2, I will analyze the above example graphically and give some other graphical examples.

-mike-