No 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,
An identity is always true, no matter what x is
A contradiction is never true for any value of x
A conditional equation is true for some values of
x
Addition Principle
Equivalent Equations
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
ex pression 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.
The Addition Principle
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 al lows 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.
Notations
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.
Multiplication Principle
Multiplying (or dividing) the same non- zero number
to both sides of an equation does not change its
solution set.
Example:
so if 6x = 12, then 18x = 36 for the same value of x
(which in this case is x = 2).
The way we use the multiplication principle to solve
equations is that it allows us to isolate the variable by
getting rid of a factor that is multiplying the variable.
Example: 2x = 6
To get rid of the 2 that is multiplying the x, we can
divide both sides of the equation by 2, or multiply by
its reciprocal (one-half).
Either divide both sides by 2:
or multiply both sides by a half:
∞ Whether you prefer to think of it as dividing by
the number or multiplying by its reciprocal is not
important, although when the coefficient is a
fraction it is easier to multiply by the reciprocal:
Example: 4/5x = 8
Multiply both sides by the reciprocal of the
coefficient, or 5/4
Using the Principles Together
Suppose you were given an equation like
2x – 3 = 5.
You will need to use the addition principle to move
the –3, and the multiplication principle to remove the
coefficient 2. Which one should you use first?
Strictly speaking, it does not matter—you will
eventually get the right answer. In practice, however,
it is usually simpler to use the addition principle first,
and then the multiplication principle. The reason for
this is that if we divide by 2 first we will turn
everything into fractions:
Given: 2x – 3 = 5
Suppose we first divide both sides by 2:
Now there is nothing wrong with doing arithmetic
with fractions, but it is not as simple as working with
whole numbers. In this example we would have to
add 3/2 to both sides of the equation to isolate the x.
It is usually more convenient, though, to use the
addition principle first:
Given: 2x – 3 = 5
Add 3 to both sides:
At this point all we need to do is divide both sides
by 2 to get x = 4.
