Finding the slope of a Linear Equation

coordinate points, standard form, slope intercept form, y intercept, slope

Linear Equations are equations that represent a straight line. A linear equation can be represented in a slope intercept form or point-slope form.  We will examine both respectively and also identify their common ground, that is, the slope that represents a rate of change in many real life situations.

The slope Intercept form:

The Slope Intercept form is represented by the equation y=mx+b.

Here, m, the coefficient on x represents the slope, and b represents the y-Intercept. The y-intercept is where the function crosses the y-axis.  

Given two points, we get two pairs of x's and y's that we can use to write the equation to a given line.  Comparing the two given x's values by reading off the first of the coordinate pairs from each given point, we label the smaller x, x₁ and the larger x, x₂.  We then represent the y values accordingly.  Thus, we get two points, (x₁, y₁), and (x₂, y₂).  From this, we can figure out what the slope "m" is by the following formula.
      (y₂ – y₁)
m=-----------
      (x₂ – x₁)

After finding this m, our next task is to find the y-intercept, or "b"  

To find b, we use one of the two given points, either (x₁, y₁), or (x₂, y₂).  Suppose we picked (x₂, y₂), then, we solve for "b" by plugging in these coordinate values into our slope intercept equation.

So, y₂= m(x₂)+b. Now, subtracting m(x₂) from both sides, we get that
b= y₂-m(x₂).  

Here's a concrete example.

Suppose we are given the points (4, 8) and (2, 0). Then, comparing the X values, we see that 2 is smaller than 4, and thus (2,0) would be our (x₁, y₁) and (4, 8) would be (x₂, y₂).  So, we use the formula for m to learn that:
      (8-0)
m=-------
      (4-2)

which is 8/2, which reduces to 4, so m=4.  So now, our incomplete equation becomes:

y=4x+b, now, we need to figure out b.  This time picking (x₁, y₁) for convenience because it has a "0", we see that 0=4(2)+b, which is 0=8+b. Here, subtracting 8 from both sides, we get that b=-8.  So our complete equation is:


y=4x-8.


Now the Point-slope form.  

The point-slope form is much easier to write since it does not involve finding b, and simple algebraic manipulation would allow us to convert the standard form into the slope intercept form if we desire to find b this way.  The point-slope form is represented by the following equation.

y–y₁=m(x–x₁)

so, given the same pair of points (4,8) and (2, 0), we see that the standard equation would be

y-0=m(x-2), which reduces to y=m(x-2) and this is it!  M here is the same m we found earlier.  As you can see, the point-slope form merely requires us to plug in points once we find m and we have identified the appropriate labeling of our coordinate points.

coordinate points, standard form, slope intercept form, y intercept, slope