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.

Start Free Trial

No credit card required