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The basic approach to finding
the solution to equations
is to change the equation
into simpler equations, but
in such a way that the solution
set of the new equation is
the same as the solution set
of the original equation.
When two equations have the
same solution set, we say
that they are equivalent.
What
we want to do when we solve
an equation is to produce
an equivalent equation that
tells us the solution directly.
Going back to our previous
example,
2x + 3 = 7,
we can say that the equation
x = 2
is an equivalent equation,
because they both have the
same solution, namely x = 2.
We need to have some way to
convert an equation like 2x + 3 = 7
into an equivalent equation
like x = 2
that tells us the solution.
We solve equations by using
methods that rearrange the
equation in a manner that
does not change the solution
set, with a goal of getting
the variable by itself on
one side of the equal sign.
Then the solution is just
the number that appears on
the other side of the equal
sign.
The methods of changing an
equation without changing
its solution set are based
on the idea that if you change
both sides of an equation
in the same way, then the
equality is preserved. Think
of an equation as a balance—whatever
complicated expression might
appear on either side of the
equation, they are really
just numbers. The equal sign
is just saying that the value
of the expression on the left
side is the same number as
the value on the right side.
Therefore, no matter how horrible
the equation may seem, it
is really just saying something
like 3 = 3.
·
Adding (or subtracting)
the same number to both sides
of an equation does not change
its solution set.
Think of
the balance analogy—if both
sides of the equation are
equal, then increasing both
sides by the same amount will
change the value of each side,
but they will still be equal.
For example, if
3 = 3,
then
3 + 2 = 3 + 2.
Consequently,
if
6 + x = 8
for
some value of x (which
in this case is x = 2),
then we can add any number
to both sides of the equation
and x = 2
will still be the solution.
If we wanted to, we could
add a 3 to both sides of the
equation, producing the equation
9 + x = 11.
As
you can see, x = 2
is still the solution. Of
course, this new equation
is no simpler than the one
we started with, and this
maneuver did not help us solve
the equation.
If
we want to solve the equation
6 + x = 8,
the
idea is to get x by
itself on one side, and so
we want to get rid of the
6 that is on the left side.
We can do this by subtracting
a 6 from both sides of the
equation (which of course
can be thought of as adding
a negative six):
6 – 6 + x = 8 – 6
or
x = 2
You
can think of this operation
as moving the 6 from one side
of the equation to the other,
which causes it to change
sign
·
The addition
principle is useful in solving
equations because it allows
us to move whole terms from
one side of the equal sign
to the other. While this is
a convenient way to think
of it, you should remember
that you are not really “moving”
the term from one side to
the other—you are really adding
(or subtracting) the term
on both sides of the equation.
In
the previous example, we wrote
the –6 in-line with the rest
of the equation. This is analogous
to writing an arithmetic subtraction
problem in one line, as in
234 – 56 = 178.
You
probably also learned to write
subtraction and addition problems
in a column format, like
We
can also use a similar notation
for the addition method with
algebraic equations.
Given
the equation
x + 3 = 2,
we want
to subtract a 3 from both
sides in order to isolate
the variable. In column format
this would look like
Here
the numbers in the second
row are negative 3’s, so we
are adding the two
rows together to produce the
bottom row.
The
advantage of the column notation
is that it makes the operation
easier to see and reduces
the chances for an error.
The disadvantage is that it
takes more space, but that
is a relatively minor disadvantage.
Which notation you prefer
to use is not important, as
long as you can follow what
you are doing and it makes
sense to you. |
