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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.
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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
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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.
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