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Why Graphs of Certain Algebraic Equations --
Those in the Form of Y = aX + b --
Come Out Linear (i.e., Straight Lines)
by Rick Garlikov

[The significant feature of these equations is that the "X" is not raised to an exponent higher than 1;
that is, you don't get straight lines when you graph the values of X and Y in equations such as
(Y = aX2 + ...)  or (Y = aX5 + ...)]


Look at a fairly simple algebra equation, say, Y = 2X + 1.
Suppose we want to graph, on a typical XY coordinate graph, the values of Y which we get for each value we might choose for X. 
If X were 1, Y would be 3; if X were 2, Y would be 5, if X were 3, Y would be 7.
We could make a table of these values and get:
If X is:
Then Y is:
0
1
1
3
2
5
3
7
4
9
5
11
6
13
Notice the pattern: every time you increase X by 1, you increase Y by 2. 
(Click here, after you read the rest of this, for a fuller, and more general, 
but more complicated-looking explanation of this point if you wish.)

That means every time you go over to the right by one amount on the X axis, you will go upwards by two on the Y axis.  (And, if you go to the left by one on the X axis, you will also  go downwards by 2 on the Y axis.)  That means there is a steady or constant incline or slope of 2 to 1, and a constant slope (by convention, the amount it goes up or down divided by the amount it goes "over" to the right or to the left) of any hill or any incline will look like (and actually be) a straight line, just as when you lay a long, flat board on a well-built set of stairs that have the same rise for every inch of depth of the step; the board will touch all the step edges at the same time, showing those edge points are in a straight line.  (I don't see why a constant rise for any given amount of distance over -- what is called "the run" -- gives a straight line, but I can see that it does.  Maybe I will see why by tomorrow.)

We can generalize then: if you have any equation of the form Y = aX + b, 
 

If X is:
Then Y is:
0
b
1
a + b
2
2a + b
3
3a + b
4
4a + b
5
5a + b
6
6a + b

the same will hold true -- increasing X by any amount will increase Y by "a" times that amount-- and the slope of the  line -- the ratio of the increase on the Y axis for any increase on the X axis -- will be "a".  And where X is zero, the point on the Y axis will be "b", since aX will equal zero, and Y therefore will equal b.

Then for any straight line, if you know where one point on it is, and if you know the angle of the line, you can figure out any of the other points by drawing the line on a graph and looking at the coodinates you are interested in.

Working this backward, from a straight line to an equation solved by the same points that line is on*, you can find the slope of the line by finding out how much it goes up or down for any distance it goes to the right (or left).  If it goes up 8 for every 1 it goes to the right, the slope or "a" in the equation will be 8.  If it goes down 8 for every 1 it goes to the right, the slope will be -8.  Then you need to find out where it crosses the Y axis, and that will be "b".   You need to remember to keep your directions and your plus and minus signs lined up correctly when you do these, the convention being that moving to the right and moving upwards are going in the positive direction, and moving downward and moving to the left are going in negative directions.

For some reason, what I have called "a" above in the equation is called "m" when talking about equations in this form.  I don't know when or why "m" became the symbol for a slope.

This work is available here free, so that those who cannot afford it can still have access to it, and so that no one has to pay before they read something that might not be what they really are seeking.  But if you find it meaningful and helpful and would like to contribute whatever easily affordable amount you feel it is worth, please do do.  I will appreciate it. The button to the right will take you to PayPal where you can make any size donation (of 25 cents or more) you wish, using either your PayPal account or a credit card without a PayPal account.

 













 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

At first it seems like it should increase by 3 because you are doubling the amount you have increased X by and then adding 1 to it.  But that is not what is happening because you had also added 1 to the previous value of (2 times X) in order to get the value for that Y.
If we look at that symbolically, and generally, using the equation
Y = aX + b
suppose we have Xn and Xn+1differing by 1.  And call the corresponding Y's then Ynand Yn+1.  If we want to find the difference between any Y and the next Y on the table where X has been increased by one, we can write that as:
Yn+1- Yn
Which will be then (by substitution):
(aXn+1 + b) - (aXn + b)
Which is:
aXn+1 - aXn

So the "b" drops out of the picture when computing 
the difference between the values of Y.

And since Xn+1 = Xn +1, we can say that:

Yn+1- Yn = a[(Xn + 1)] - (aXn)]
Which is:
(aXn + a) - (aXn)
Which is:
a

So therefore the difference between the value of Y for any whole number value of X and the value of Y for the next whole number value of X will be "a", and increasing the value of X by any number will increase the value of Y by that number times "a". (Return to text.)

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

* I say that the line is on, or runs through, the same points that solve the equation because that seems to me to be the most accurate characterization of the relationship between the line and the equation.  I don't think that the line "represents" the equation, or vice versa, in any other sense; and certainly the line and the equation are not the same thing.  And it is misleading to say that the equation is "the equation of the line".  The relationship between the line and the equation is complicated. It results from a mathematical construct in the sense that you have to go through a series of prescribed steps in order to get this line.  That is you have to describe a coordinate system and explain what the axes mean, how to plot points, etc.  Then you have to describe how to translate values in the equation for values on the coordinate system.  Then plot the points and connect them.  That is when you get a straight line.  If you change any of this "recipe" for making a graph, you may not get a straight line.  So the straight line is related to this whole procedure, not just to the equation by itself.  And you get different kinds of lines (curved ones of various sorts, and other kinds as well) when you have equations where the "x" is raised to some exponent or "power" higher than 1.

When you were kids drawing lines with rulers, you weren't doing it in terms of equations, so equations have nothing to do with lines in any conscious or natural sense.  But by constructing things in certain ways mathematically, or by a certain recipe, it turns out that relationships can be discovered between lines and equations.  Machines and computer software can then be developed to draw lines, etc., but that is not how humans draw lines, or what they mean by lines, any more than humans learn to play tennis by computing trajectories using calculus and Newton's laws of motion along with knowledge of the acceleration caused by the force of gravity.  People have figured out how to relate mathematical things to things in the real world, even though those things in the real world are not the same as the mathematical things.  If we could make a robot that could run and hit a tennis ball, using radar and math, it would probably beat us at tennis all day long, but it would not be playing tennis the same way we would be (and would probably not even know what tennis is).  It is interesting that we can use math to do things better than we do them normally, even though doing them normally has nothing to do with math. (Return to text.)