Basic Math and Pre-Algebra For Dummies (94 page)

Knowing how to evaluate arithmetic expressions comes in handy for evaluating algebraic expressions. For example, suppose you want to evaluate the following expression:

• 4
  x
  – 7
  

Note that this expression contains the variable
x,
which is unknown, so the value of the whole expression is also unknown.

An algebraic expression can have any number of variables, but you usually don't work with expressions that have more than two or maybe three, at the most. You can use any letter as a variable, but
x, y,
and
z
tend to get a lot of mileage.

Suppose in this case that
x
= 2. To evaluate the expression, substitute 2 for
x
everywhere it appears in the expression:

• 4(2) – 7
  

After you make the substitution, you're left with an arithmetic expression, so you can finish your calculations to evaluate the expression:

• = 8 – 7 = 1
  

So given
x
= 2, the algebraic expression 4
x
– 7 = 1.

Now suppose you want to evaluate the following expression, where
x
= 4:

Again, the first step is to substitute 4 for
x
everywhere this variable appears in the expression:

Now evaluate according to the order of operations explained in Chapter
5
. You do powers first, so begin by evaluating the exponent 4
²
, which equals 4 × 4:

Now proceed to the multiplication, moving from left to right:

Then evaluate the subtraction, again from left to right:

So given
x
= 4, the algebraic expression 2
x
²
– 5
x
– 15 = –3.

You aren't limited to expressions of only one variable when using substitution. As long as you know the value of every variable in the expression, you can evaluate algebraic expressions with any number of variables. For example, suppose you want to evaluate this expression:

To evaluate it, you need the values of all three variables:

The first step is to substitute the equivalent value for each of the three variables wherever you find them:

Now use the rules for order of operations from Chapter
5
. Begin by evaluating the exponent 3
²
:

Next, evaluate the multiplication from left to right (if you need to know more about the rules for multiplying negative numbers, check out Chapter
4
):

• = 27 + (–12) – (–30)
  

Now all that's left is addition and subtraction. Evaluate from left to right, remembering the rules for adding and subtracting negative numbers in Chapter
4
:

• = 15 – (–30) = 15 + 30 = 45
  

So given the three values for
x
,
y
, and
z
, the algebraic expression 3
x
²
+ 2
xy
–
xyz
= 45.

 For practice, copy this expression and the three values on a separate piece of paper, close the book, and see whether you can substitute and evaluate on your own to get the same answer.

 A
term
in an algebraic expression is any chunk of symbols set off from the rest of the expression by either addition or subtraction. As algebraic expressions get more complex, they begin to string themselves out in more terms. Here are some examples: