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Fearless Symmetry by Robert Gross, Avner Ash

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56 CHAPTER 6
Systems of Equations
Next,wewanttoconsidersystems of equations. In English, we often
connect our assertions with “and”: The first president of the United
States was George Washington and the sixteenth president was
Abraham Lincoln. We are asserting that both statements are true.
We can do this with unknowns as well. Consider:
1. X = the first president of the United States.
2. Y = the sixteenth president of the United States.
This system has the solution X=GeorgeWashington,Y=Abraham
Lincoln. Solving the system means solving both equations at once.
In this example, the equations do not share any unknowns, so
solving both is the same as solving each one separately. W e say
that the equations are uncoupled. That is not very interesting.
Consider now:
1. X = the first president of the United States.
2. X = the sixteenth president of the United States.
Now the equations are coupled, and there is no solution to the
system, although each equation can be solved separately. (By the
way, there is a pair of numbers that can be put in place of “first”
and “sixteenth” for which the system has the unique solution X =
Grover Cleveland.) Finally, consider:
1. X = the first president of the United States.
2. X = the commanding officer of the colonists during the
American Revolution.
This coupled system does ha ve a solution, again unique, namely,
X = George Washington.
Of course, solution sets to systems of equations need not contain
only one element. To put it another way, solutions need not be
unique. For a given system of equations, there may be no solutions,
exactly one, more than one but still finitely many, or infinitely many
solutions. For example, consider the system of Z-equations:
1. x + y = z.
2. x
2
+ 2xy + y
2
= z
2
.
EQUATIONS AND VARIETIES 57
Any solution to the first equation is also a solution to second
equation. Because any three numbers for which the sum of the first
two is the third gives a solution to the first equation (hence to both),
we see there are infinitely many solutions to the two equations,
even if we allow only integer solutions.
Here are some more examples of systems of equations and the
varieties they define. First, a nonexample:
1. x
2
+ y
2
= 1.
2. x > 0.
This system makes perfect sense, but we do not allow it because it
involves an inequality in the second statement. Inequalities have
no meaning when applied to some number systems, for instance C
or F
p
. So this system does not define an algebraic variety.
Next, an example with more than two equations in the system:
1. x
2
+ y
2
+ z
2
= w.
2. w
4
= 1.
3. x + y = z.
J ust for fun, let us find S(Z) for this system. From (2) we see
that w = 1orw =−1. There are no solutions to (1) if w =−1,
and if w = 1, we get that x, y and z are all 0 or ±1, and exactly
two of them equal 0. But then (3) is impossible. So S(Z)isthe
empty set.
Now, we find S(R) for this system. Again, from (2) we get that
w = 1orw =−1, and again equation (1) has no solutions when
w =−1. We now substitute x + y for z in (1), which we are allowed
to do because of (3). Thus, S(R) is the same as the set of R-solutions
of the following single equation:
4. x
2
+ y
2
+ (x + y)
2
= 1.
or equivalently
5. x
2
+ y
2
+ xy = 1/2.
The solutions to (5) may be graphed on (x, y)-graph paper as an
ellipse. There are infinitely many R-solutions to (5), and so S(R)is
infinite.

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