Question 143403
First of all I don't think you meant to say "an equation."  I think you meant "a system of equations."


Every equation in two variables has a corresponding straight line graph in the coordinate plane.  Therefore, a system of two equations represents two lines in the plane, and there are three possible situations:


1.  The two lines intersect in a single point.


2.  The two lines are parallel and do not intersect at all.


3.  The two lines are actually the same line and intersect in every point on the line.


The solution set of a system of equations is the set of ordered pairs (points on the plane) that satisfy both equations simultaneously.  The coordinates of a point satisfy both equations if and only if the coordinates form an ordered pair that represents a point of intersection of the two graphs represented by the equations.


So, for situation 1 above, you have exactly 1 ordered pair in the solution set.  For situation 2, you have an empty solution set, i.e. no solutions.  And for situation 3, you have a solution set with an infinite number of elements.


In algebraic terms, given two equations in two variables and you use the elimination method to solve the system you will obtain one of three results:


1.  You will get a single pair of values representing the two variables (Situation 1 above)


2.  Your equations will reduce to an absurdity, something like {{{6=0}}}, meaning that there is no solution (Situation 2 above), or


3.  Your equations will reduce to a trivial identity, something like {{{0=0}}}, meaning that there are an infinite number of solutions (Situation 3 above)


Example:
Situation 1:
{{{x-y=2}}}
{{{x+y=6}}}


Add the two equations:
{{{2x+0y=8}}}
{{{x=4}}}
{{{4-y=2}}}
{{{y=2}}}


Solution set: {(4,2)}


Situation 2:
{{{x-y=2}}}
{{{2x-2y=6}}}

Multiply the first equation by -2 and add the result to the 2nd equation:
{{{-2x+2y=-4}}}
{{{2x-2y=6}}}


{{{0x+0y=2}}}
{{{0=2}}}.  Absurd result, therefore no solution.


Situation 3:
{{{x-y=2}}}
{{{2x-2y=4}}}


Multiply the first equation by -2 and add the result to the 2nd equation:
{{{-2x+2y=-4}}}
{{{2x-2y=4}}}


{{{0x+0y=0}}}
{{{0=0}}}.  Trivial identity, therefore infinite solutions.


Here's another, perhaps simpler way to do it.


Put both of your equations into slope-intercept form by solving for y ({{{y=mx+b}}}), remembering to reduce to lowest terms.


1.  If the slopes (m) are different, then you have situation 1, a single solution.


2.  If the slopes are the same but the intercepts are the different, then you have situation 2, no solution -- they are different but parallel lines.


3.  If the slopes are the same AND the intercepts are the same, then you have situation 3, infinite solutions -- they are the same line.