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### Equations with Two Variables

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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

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,

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.

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.

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

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/
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Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations

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