To solve an equation, you have to isolate the variable
you’re solving for. You have to “manipulate” the equation until
you get the variable alone on one side of the equal sign. By definition,
the variable is then equal to everything on the other side of the equal
sign. You’ve just solved for the variable. Congratulations.
The Fine Art of Manipulation
You can’t manipulate an equation the way you manipulate
your little brother or sister. When manipulating equations, there
are rules. Here’s the first and most fundamental (it’s so important
we’re going to bold it): Whatever you do to one side of an equation,
you must do to the other side. If you divide one side of
an equation by 3, 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.
If you fall in love with one side of the equation, fall in love with
the other. Neither side will think you’re a two-timer. They’ll think
you’re a highly skilled mathematician.
By treating the two sides of the equation in the same
way, you don’t change what the equation means. You change the form of
the equation—that’s the point of manipulating it—but the equation
remains true since both sides stay equal.
Take, for instance, the equation 3x +
2 = 5. You can do anything you want to it, anything at all,
and as long as you do that thing to both sides of the equation, x will always
equal 1. For example, if you subtract 2 from both sides,
3x + 2 – 2 = 5 – 2
3x = 3
x = 1
And if you multiply both sides by 2,
2(3x + 2) = 2(5)
6x + 4 = 10
6x = 6
x = 1
In addition to the “do the same things to both sides of
the equation” rule that you must follow, there are other rules of
manipulation that you should follow. Nothing will
go horribly wrong if you don’t follow them, but it will take you
longer to solve the question. Here are the rules:
Combine like terms to make the equation simpler.
the equation in the reverse order of operations.
The second rule means that you should first subtract or
add any extra terms on the same side as the variable. Then divide
and multiply anything on the same side as the variable. Next, raise
both sides of the equation to a power or take their roots according
to any exponent attached to the variable. Finally, work out anything
inside parentheses. Do the order of operations backward: SADMEP!
The idea is to “undo” everything that has been done to the variable
so that it will be isolated in the end. Example time:
In this equation, poor little x is
being square rooted, multiplied by 2, added to 3,
and encased in parentheses. You’ve got to get him out of there!
Undo all of these operations in order to liberate x.
First, subtract 2 from both sides of the
Then multiply both sides by 2 to get rid
of the fraction:
Now divide both sides by 3 (later, parentheses):
Subtract 3 from each side:
Divide both sides by 2:
And, finally, square each side to get rid of the square
Success! You’ve freed x from
all of those bullying operations.
Location. Location. Location.
Isolating for x is all about
where x is located. A variable in
the numerator of a fraction is actually a pretty easy location to
isolate. But if x is in the denominator
of a fraction, things get more complicated.
Following SADMEP, start by subtracting the 3:
But now you have to get the x out
of the denominator, and the only way to do that is to multiply both
sides of the equation by that denominator, x +
Divide both sides by 4:
Subtract 2 from each side:
By now you know the rule: When solving an equation, never
do something to one side of an equation that you don’t do to the
other. If you add 4 to one side, you have to add 4 to the other.
But what if there were some simplification tools that didn’t change
the value of an expression? What if you could simplify one side
of an equation without changing its value? That would rock. Why?
Because it would allow you to make solving equations much simpler
and save you time on the SAT.
The first step to adding ferocious simplification tools
to your arsenal is the rule of distribution, which states
a can be any kind of term, meaning it
could be a variable, a constant, or a combination of the two.
When you distribute a factor into an expression within
parentheses, multiply each term inside the parentheses by the factor
outside the parentheses. For example, in the previous problem, when
you had 1 = 4(x + 2), you didn’t actually
have to divide both sides by 4. You could have distributed the 4
and pushed off all those messy fractions until the end: 1
= 4x + 8; 4x = –7; x =
So, if you have the expression 3y(y2 –
Seems logical enough. But the true value of distributing
becomes clear when you see a distributable expression in an equation: 3y(y2 +
6) = 3y3 + 36 looks
like it’d be hard to solve, since there aren’t any equal terms to
add or subtract away. But wait a sec . . . what if you distribute
Shocking revelation! It’s suddenly clear that you can
subtract 3y3 from
Factoring an expression is the opposite of distributing. 4x3 –
8x2 is one mean-looking expression.
Or so it seems, until you realize that all the terms share the greatest
common factor 4x2,
which you can factor out:
With distributing and factoring, you can group or ungroup
quantities in an equation to make your calculations simpler. Here
are a few more examples:
Combining Like Terms
After factoring and distributing, you can take additional
steps to simplify expressions or equations. Combining like terms
is one of the simplest techniques you can use. It 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 by adding the coefficients of the variable x3 (–1 and 3)
together and the coefficients of x2 (1 and 4)
Variables that have different exponential
values are not like terms and can’t be combined.
Two terms that do not share a variable are also not like terms and
cannot be combined regardless of their exponential value.