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Equations with Two Variables


Date: 03/25/98 at 00:00:15
From: Gina M
Subject: Algebra 1

The subject is solving systems of equations by graphing. I need to 
learn how to identify solutions, and how to find solutions by 
graphing. Examples:

   1)  Determine whether (1,2) is a solution of the system.

              y  =  x + 1
         2x + y  =  4

   2)  Solve by graphing.
  
         x + 2y  =  7
              x  =  y + 4

Then I have inequalities in 2 variables. I need to know solutions of 
inequalities in 2 variables and graphing equations in 2 variables.

Examples:

   1)  Determine wheather (5, -3) is a solution of the inequality 

       2x - y  >  5.

   2)  Graph  x = y < 4  on a coordinate plane.

Then there is graphing systems of linear inequalities. Example:

   Solve this system by graphing.

       2x + y  <=  8
       2x - y  >   1

   I know you have to use plug-in's and shading or something.

I bet I sound really confused, so if you can make out what I'm trying 
to ask, please please help.

Thank you,
Gina


Date: 03/25/98 at 16:38:34
From: Doctor Rob
Subject: Re: Algebra 1

You can tell whether a pair (a,b) is a solution of a system of 
equations in two variables x and y as follows.  

Wherever you see an "x" in the equations, write the value of a, and 
wherever you see a "y" in the equations, write the value of b.  
(This is called "substituting a for x" and "substituting b for y".  
More informally, it is called "plugging in a for x" and "plugging in 
b for y".)  Simplify the results, and see if the equations you have 
are true.  If all are true, then the pair (a,b) is a solution, and 
if it is false, then the pair (a,b) is not a solution.

That should take care of your first problem:

   Determine whether (1,2) is a solution of the system.

         y = x + 1
   2*x + y = 4

Substituting, you get 

         2 = 1 + 1 and 
   2*1 + 2 = 4

Are these equations true? Yes, both are true, so the pair (1,2) is, 
indeed, a solution of the given system of equations.

All the pairs (a,b) that satisfy one equation can be plotted on a 
graph. The value of a is called the x-coordinate and the value of b 
is called the y-coordinate of the point. The point P with coordinates 
(a,b) is  a  units to the left of the y-axis (if a is positive) or -a 
units to the right of it (if a is negative).  P is also b units above 
the x-axis (if b is positive) or -b units below it (if b is negative).

Of course you can't plot *all* the pairs (a,b), since there are 
infinitely many of them! But you can plot some, and use the knowledge 
that you will have to have a straight line if x and y appear only 
to the first power in the equation. Once you have 3 or 4 points 
plotted, you should be able to draw the line quite easily, either with 
a ruler or freehand. If the points don't seem to make a straight line, 
check your arithmetic!

To find the solutions of two linear equations, plot the lines that are 
the graphs of the two equations. For a pair (a,b) to make both 
equations true, it must be on both lines. Since two lines intersect 
in a single point (if they intersect at all), there will be a single 
point on both lines. You can estimate the coordinates (a,b) of that 
point P, and you will have guessed the simultaneous solution to the 
two equations. Of course you will want to check that your guess 
actually does satisfy the two equations.

That should do it for your second problem:

   Solve by graphing.

     x + 2*y  =  7
           x  =  y + 4

For the first equation, pick a few values of y and compute the value 
of x that corresponds:  y = 0, x = 7;  y = 1, x = 5;  y = 3, x = 1;  
y = 4, x = -1. Now plot the points (7,0), (5,1), (1,3), and (-1,4), 
and draw the line through them. Do the same for the second equation:  
(2,-2), (4,0), (6,2), (8,4).  Now see if you can figure out the 
coordinates of the point where the two lines intersect.

The same ideas work for inequalities, with this exception:  

  The points that satisfy a linear inequality all lie on one side 
  of the line you have by replacing the inequality sign (>, <, >=, <=) 
  by an equal sign. This is usually depicted on a graph by drawing 
  the boundary line and shading in the half of the plane on the  
  correct side of that line.  If the inequality is strict (< or >), 
  points on the line are not included in the set of solutions, but 
  if it is not (<= or >=), they are included.

Your third problem is this:

   Determine wheather (5,-3) is a solution of the inequality 

     2*x - y > 5

Substitute, getting 2*5 - (-3) > 5.  Is this true?  If so, (5,-3) 
is a solution, and if not, then not.

Now to solve two linear inequalities, draw the two solution sets 
(with shading), and find the part of the plane that is the overlap 
(or intersection) of the two solution sets.

Here is your last problem:

   Solve this system by graphing.

   2*x + y <= 8,
   2*x - y >  1.

Graph the boundary lines 

   2*x + y  = 8    and 
   2*x - y  = 1,   using the technique above.  

The solution set of the first inequality is the first line plus
everything on the same side of that line as (0,0) [since (0,0) 
satisfies the first inequality]. The line is included since the 
inequality sign includes the possibility of equality.

The solution set of the second inequality is everything on the 
opposite side of the second line from (0,0) [since (0,0) doesn't 
satisfy the second inequality].  

The overlap of these two regions is the solution set of the 
simultaneous system of two inequalities in two unknowns you 
were given.

-Doctor Rob,  The Math Forum
Check out our web site http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations

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