## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

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.

## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

No credit card required