Now that you know how to set up the equation, the next
thing to do is to solve for the value that the question asks for.
First and foremost, the most important thing to remember when manipulating
equations is to do exactly the same thing to each side of the equation.
If you divide one side of an equation by 3, you must divide the
other side by 3. If you take the square root of one side of an equation,
take the square root of the other.
By treating the two sides of the equation in the same
way, you can rest easy that you won’t change the meaning of the
equation. You will, of course, change the form of
the equation—that’s the point of manipulating it. But the equation
will always remain true as long as you always do the same thing
to both sides.
For example, let’s look at what happens when you manipulate
the equation 3x + 2 = 5, with x =
Subtract 2 from both sides:
Multiply both sides by 2:
These examples show that you can tamper with the equation
in any way you want, as long as you commit the same tampering on
both sides. If you follow this rule, you can manipulate the question
how you want without affecting the value of its variables.
Solving an Equation with One Variable
To solve an equation with one variable, you must isolate
that variable. Isolating a variable means manipulating the equation
until the variable is the only thing remaining on one side of the
equation. Then, by definition, that variable is equal to everything
on the other side, and you have successfully “solved for the variable.”
For the quickest results, take the equation apart in the
reverse order of operations. That is, first add and subtract any
extra terms on the same side as the variable. Then, multiply and
divide anything on the same side of the variable. Next, raise both
sides of the equation to a power or take their roots according to
any exponent attached to the variable. And finally, do anything
inside parentheses. This process is PEMDAS in reverse (SADMEP!). The
idea is to “undo” everything that is being done to the variable
so that it will be isolated in the end. Let’s look at an example:
In this equation, the variable x is
being squared, multiplied by 3, added to 5, etc. We need to do the
opposite of all these operations in order to isolate x and
thus solve the equation.
First, subtract 1 from both sides of the equation:
Then, multiply both sides of the equation by 4:
Next, divide both sides of the equation by 3:
Now, subtract 5 from both sides of the equation:
Again, divide both sides of the equation by 3:
Finally, take the square root of each side of the equation:
We have isolated x to show that x =
Sometimes the variable that needs to be isolated is not
conveniently located. For example, it might be in a denominator
or an exponent. Equations like these are solved the same way as
any other equation, except that you may need different techniques
to isolate the variable. Let’s look at a couple of examples:
for x in the equation + 2 = 4.
The key step is to multiply both sides by x to
extract the variable from the denominator. It is not at all uncommon
to have to move the variable from side to side in order to isolate
Remember, performing an operation on a variable is mathematically
no different than performing that operation on a constant or any
Here’s another, slightly more complicated example:
This question is a good example of how it’s not always
simple to isolate a variable. (Don’t worry about the logarithm in
this problem—we’ll review these later on in the chapter.) However,
as you can see, even the thorniest problems can be solved systematically—as long
as you have the right tools. In the next section, we’ll discuss
factoring and distributing, two techniques that were used in this
So, having just given you a very basic introduction to
solving equations, we’ll reemphasize two things:
Do the same thing to both sides.
backward (with respect to the order of operations).
Now we get into some more interesting tools you
will need to solve certain equations.
Distributing and Factoring
Distributing and factoring are two of the most important
techniques in algebra. They give you ways of manipulating expressions
without changing the expression’s value. So it follows that you
can factor or distribute one side of the equation without doing
the same for the other side of the equation.
The basis for both techniques is the following property,
called the distributive property:
a can be any kind of term, from a variable
to a constant to a combination of the two.
When you distribute a factor into an expression within
parentheses, you simply multiply each term inside the parentheses
by the factor outside the parentheses. For example, consider the
expression 3y(y2 –
If we set the original, undistributed expression equal
to another expression, you can see why distributing facilitates
the solving of some equations. Solving 3y (y2 –
6) = 3y3 + 36 looks
quite difficult. But if you distribute the 3y,
Subtracting 3y3 from
both sides gives us:
Factoring an expression is essentially the opposite
of distributing. Consider the expression 4x3 –
8x2 + 4x,
for example. You can factor out the GCF of the terms, which is 4x:
The expression simplifies further:
See how useful these techniques are? You can
group or ungroup quantities in an equation to make your calculations
easier. In the last example from the previous section on manipulating equations,
we distributed and factored to solve an equation.
First, we distributed the quantity log 3 into the sum of x and
2 (on the right side of the equation). We later factored the term x out of
the expression x log 2 – x log
3 (on the left side of the equation).
Distributing eliminates parentheses, and factoring creates
them. It’s your job as a Math IC mathematician to decide which technique
will best help you solve a problem.
Let’s see a few examples:
Combining Like Terms
After factoring and distributing, there are additional
steps you can take to simplify expressions or equations. Combining
like terms is one of the simpler techniques you can use, and involves
adding or subtracting the coefficients of variables that are raised
to the same power. For example, by combining like terms, the expression:
can be simplified to:
by adding the coefficients of the variable x3 together
and the coefficients of x2 together.
Generally speaking, when you have an expression in which
one variable is raised to the same power in different terms, you
can factor out the variable and add or subtract the coefficients,
combining them into one coefficient and therefore combining the
“like” terms into one term. A general formula for combining like
pairs looks something like this: