Equations of the Line

••• For pre-algebra students, focus on the slope intercept form; the others will be addressed in Algebra 1 and, perhaps, Algebra 2.

                    AY + BX + C = 0                “general”

 

                             “intercept”

                    y = mx + b                       “slope intercept”

                                   “point slope”

Linear equations have variables whose highest power is 1.  Also the variables are not multiplied together.

The above three forms are equations of a line.

We say there is a linear relationship between the variables in these equations. This linear relationship implies that as one variable changes the other changes with a constant rate of change.

For instance, say I work at a job that pays $40 per hour, then for each hour I work I should get another $40, not $50, not $10, etc., but a constant $40. Say I worked 10 hours. Then I should get paid $400. Suppose further that I initially was paid $100 when I showed up to work for the first time as a bonus. Then in this case, after working 10 hours I will have made the initial $100 + the earned $400 which is $500. The following linear equation models my pay:

pay = $40 ● hours_worked + $100

y = m ● x + b

m = $40 = pay rate of change.

b = $100 = initial amount at start.

pay = y

hours_worked = x

In this example, the amount I get paid “depends” on the amount of hours I work.

To capture this meaning, we say the y depends on x, that is, y is the dependent variable and x is the independent variable. The following graph illustrates this relationship.

The general equation is useful studying equations in general. By adding variables and powers of variables one can see the differences between the conic sections (covered later).

The intercept form and the slope intercept forms of the equation of a line are useful when graphing the lines or determining equation from the graph of a line.

Consider the graph below:

We can see that the x intercept is -3 and that the y intercept is 2 so the

equation of this line using the intercept form is:

                                  

The point slope form of this equation requires a slope 'm'. Recall that m is the change in y divided by the change in x. It is helpful to list the points that are used to find this slope. The points I will use are (3,4) and (6,6) shown on the graph above. Moving from the first point to the next the change in x is 3 (positive 3 since we are moving in the positive x direction). The change in y is 2, again moving vertically in a positive y direction. The slope m is then 2 / 3.

So using one of the points the point slope equation of the line is:

                            

The slope intercept form of this equation is (recall b is they intercept)

                           

The general form of this equation can be determined with the previous equation like so (multiplying throughout by 3):

                  

            so:       A = 2, B = -3, and C = 6

Now we need to show these equations are equivalent.

       

          

        

So in this case, the intercept form is the easiest to use for the equation of this line since both intercepts can readily be read right from the graph.

The other two equations requires the slope 'm' to be known then also the intercept for the slope intercept form or a point on the line for the point slope form.

Many texts will introduce the point slope form of an equation of a line as:

                                   y – y₁ = m(x – x₁)

Doing this only confuses the student more than is necessary. And here's why:

                                   the slope 'm' is the

'change in one value compared to the change of another value.' And furthermore this comparison is stated as a ratio of the changes and this ratio is 'constant.' If the ratio is not constant we do not have a line.

We are comparing the change in y to the change in x like so:

                                       

Well this can also be written as

                                        

which is used in calculus to define the derivative of Y with respect to X by allowing ∆X to approach 0. Applied to the equation of a line, this is still the slope 'm' of that line. When applied to a curve at a point this can be interpreted as the slope of the tangent to the curve at that point.

With lines, curves, and surfaces, we're still talking about a slope (or tangent), and beyond 3 dimensions we're still talking about a rate of change.

The point is, teaching the point slope form the way I suggest will reinforce

                            the importance of 'slope'

and more importantly the ratio of the change of one variable to the change of another variable. The concept of a derivative will then become natural later to the student in Calculus.

So what is the point of equations of a line and its graph?

1. From the graph that the dependent variable depends on the independent variable in a constant way, this constant way is the slope of the graph. And, this relationship is linear.

2. Given a point and a slope, two intercepts, a point and an intercept, or two points, the equation of that line can be determined.

    

   
   This is another FREE Algebra PRINTABLE presented to you from the Algebra section of K12math.com