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Up
until now, we have just been
talking about manipulating
algebraic expressions. Now
it is time to talk about equations.
An expression is just a statement
like
2x + 3
This expression might be
equal to any number, depending
on the choice of x.
For example, if x = 3
then the value of this expression
is 9. But if we are writing
an equation, then we are making
a statement about its value.
We might say
2x + 3 = 7
A mathematical equation is
either true or false. This
equation, 2x + 3 = 7,
might be true or it might
be false; it depends on the
value chosen for x.
We call such equations conditional,
because their truth depends
on choosing the correct value
for x. If I choose
x = 3, then
the equation is clearly false
because 2(3) + 3 = 9,
not 7. In fact, it is only
true if I choose x = 2.
Any other value for x
produces a false equation.
We say that x = 2
is the solution of
this equation.
- The
solution of an equation
is the value(s) of the variable(s)
that make the equation a
true statement.
An equation like 2x + 3 = 7
is a simple type called a
linear equation in one variable.
These will always have one
solution, no solutions, or
an infinite number of solutions.
There are other types of equations,
however, that can have several
solutions. For example, the
equation
x2 = 9
is satisfied by both 3 and
–3, and so it has two solutions.
This is the normal case,
as in our example where the
equation 2x + 3 = 7
had exactly one solution,
namely x = 2.
The other two cases, no solution
and an infinite number of
solutions, are the oddball
cases that you don’t expect
to run into very often. Nevertheless,
it is important to know that
they can happen in case you
do encounter one of these
situations.
Consider the equation
x = x
This
equation is obviously true
for every possible value of
x. This is, of course,
a ridiculously simple example,
but it makes the point. Equations
that have this property are
called identities.
Some examples of identities
would be
2x = x + x
3 = 3
(x – 2)(x + 2) = x2
– 4
All
of these equations are true
for any value of x.
The second example, 3 = 3,
is interesting because it
does not even contain an
x, so obviously its truthfulness
cannot depend on the value
of x! When you are
attempting to solve an equation
algebraically and you end
up with an obvious identity
(like 3 = 3), then
you know that the original
equation must also be an identity,
and therefore it has an infinite
number of solutions.
Now
consider the equation
x + 4 = x + 3
There
is no possible value for x
that could make this true.
If you take a number and add
4 to it, it will never be
the same as if you take the
same number and add 3 to it.
Such an equation is called
a contradiction, because
it cannot ever be true.
If you
are attempting to solve such
an equation, you will end
up with an extremely obvious
contradiction such as 1 = 2.
This indicates that the original
equation is a contradiction,
and has no solution.
In summary,
o
An
identity is always
true, no matter what x
is
o
A
contradiction is never
true for any value of x
o
A conditional
equation is true for some
values of x |
