Graphing Slanted Lines

Date: 07/22/97 at 16:05:47
From: P. J.  Richardson
Subject: Graphing slanted lines

When graphing slanted lines what does it mean when the instructions 
say add, subtract, or divide both sides by a particular number? And 
how do you know when to add, subtract, or divide?  Please give an 
example.  

Regards, P.J.

Date: 07/28/97 at 12:22:03
From: Doctor Rob
Subject: Re: Graphing slanted lines

Suppose you had the equation 2*x - 3*y + 5 = 0. Since it has degree 1, 
it is the equation of a line. As an equation, it has two sides: the 
left, 2*x - 3*y + 5, and the right, 0.  

You are going to try to solve this equation for y. The way to do that 
is to move terms from one side to the other by adding something to 
both sides. For example, to move the 5 from the left side to the 
right, add -5 to both sides. Since -5 = -5, and equals added to equals 
are equal, you get

   (2*x - 3*y + 5) + (-5) = 0 + (-5)

or, in other words,

   2*x - 3*y = -5.

Move the 2*x term to the other side by adding -2*x = -2*x to this last
equation:

   2*x - 3*y + (-2*x) = -5 + (-2*x),
or

   -3*y = -2*x - 5.

We have isolated y on one side of the equation, and everything else on
the other side, which is desirable. Now we are going to divide this
equation by -3 = -3. Since equals divided by nonzero equals are equal,
we will get a true equation:

   (-3*y)/(-3) = (-2*x - 5)/(-3)

or

   y = (2/3)*x + (5/3).

This completes the process of solving for y. We solved for y to put
the equation into the slope-intercept form, y = m*x + b.

In this form, y = m*x + b, you can read off m = 2/3 and b = 5/3.  
m is the slope of the line, and b is the y-intercept, that is, the 
point (0,b) on the line and also on the y-axis. Now you can graph the 
line quite easily.

Another method is to rewrite the equation in the form x/a + y/b = 1.
To do this, move all the constant terms to the right side, and all the
x- and y-terms to the left side. Divide both sides by the constant on
the right to make it equal to 1. Gather like terms, then set a to be
the reciprocal of the x-coefficient and b the reciprocal of the
y-coefficient.  In this case, we are part way there with the equation

   2*x - 3*y = -5.

Divide both sides by -5 (i.e. divide by the equation -5 = -5). Then

   (2*x - 3*y)/(-5) = (-5)/(-5)

or

   (-2/5)*x + (3/5)*y = 1

or

   x/(-5/2) + y/(5/3) = 1.

The right side is 1 as desired for this form of the equation of the
line.  From this form you can read off a = -5/2 and b = 5/3. These
are the x-intercept (a,0) and y-intercept (0,b) of the line. From
these two points it is easy to draw the line.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/