Solving Systems of Equations Graphically and Algebraically

Date: 05/01/98 at 21:55:17
From: Audra Crystal Patterson
Subject: Algebra one(Graphing Systems of Equations) 

I really do not understand how to work these problems:

   y = -x
   y = 2x

I have to graph each system of equations, then tell whether the system 
has one solution, no solution, or infinitely many solutions. If the 
system has one solution, name it. If you could please try and tell me 
and give examples, maybe it would help me understand them better. 
Thank you.
                                       
Audra Crystal Patterson

Date: 05/04/98 at 11:27:16
From: Doctor Loni
Subject: Re: Algebra one (Graphing Systems of Equations) 

Let's see if I can help you at least a little bit. I will have to make 
a few assumptions about what you have learned. (I will let you know 
what I am assuming as we go along.) Each of the equations in your 
example is a linear equation; in other words, the equation describes a 
line. I am going to assume you know how to graph points and that you 
know a little bit about lines.

If the two lines cross at one point, there is one solution to the 
equations, and the solution is the x and y values of the point where 
they cross (or intersect). If the two lines are parallel (the two 
yellow lines down a highway are parallel lines -- they are always the 
same distance apart), there is no solution, because they will never 
cross. There are infinitely many solutions if the two equations define 
the same line, which means they have every point in common.

Lines can be written in a form like this:

        y = mx + b   

where m and b are constants. m is the slope of the line, and b is the 
point where the line crosses the y axis, also known as the y-
intercept. Just like the slope of a hill, the slope of a line is the 
degree of slant. If two lines are parallel to each other, they will 
have the same slope. So if you have two equations like this:

   y = 2x + 6   and
   y = 2x + 1   

there will be no solution, because they have the same slope and will 
never cross.

If two lines are not parallel, there are two ways to find the point at 
which they intersect (which is the solution to the two equations):

   1) You can do it graphically. If your equation is y = 3x, you could
      put in values for x, solve for y, and then graph the points. For
      instance, if x is 2, y would equal 6. You could graph that
      point. Then pick at least one other value for x, find y, and
      graph that point. Connect the dots, and you have a graph of your
      line. Then you would graph the second equation in the same way,
      and where they intersect would be the solution.  

   2) You can also find the point of intersection algebraically. For 
      example, say your two equations are:

         y = 2x + 6    and
         y = -4x

      You can use the substitution method, where you solve one
      equation for one variable, and plug in what you get into the
      other equation. For instance, you already have:

         y = -4x    

      You can put -4x in for y in the other equation:
    
         y = 2x + 6

      - 4x = 2x + 6

      Solve for x:

         x = -1

      Now plug your value for x into one of the equations and solve   
      for y:

         y = -4x
         y = -4(-1) = 4

      So the solution to the two equations is x = -1 and y = 4; or, 
      written as an ordered pair, (-1,4). That is the point where
      those two lines would intersect.

Hope this is helpful. Let us know if you need more clarification.

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