Basic Math and Pre-Algebra For Dummies (102 page)

As in the preceding example, start out by removing both sets of parentheses. This time, however, on the left side of the equation, you have no sign between 3 and (–3
x
+ 1). But again, you can put your skills from Chapter
21
to use. To remove the parentheses, multiply 3 by both terms inside the parentheses:

• 11 – 9
  x
  + 3 = 25 – (7
  x
  – 3) – 12
  

On the right side, the parentheses begin with a minus sign, so remove the parentheses by changing all the signs inside the parentheses:

• 11 – 9
  x
  + 3 = 25 – 7
  x
  + 3 – 12
  

Now you're ready to isolate the
x
terms. I do this in one step, but take as many steps as you want:

• –9
  x
  + 7
  x
  = 25 + 3 – 12 – 11 – 3
  

At this point, you can combine like terms:

• –2
  x
  = 2
  

To finish, divide both sides by –2:

• x
  = –1
  

Copy this example, and work through it a few times with the book closed.

In algebra, cross-multiplication helps to simplify equations by removing unwanted fractions (and, honestly, when are fractions ever wanted?). As I discuss in Chapter
9
, you can use cross-multiplication to find out whether two fractions are equal. You can use this same idea to solve algebra equations with fractions, like this one:

This equation looks hairy. You can't do the division or cancel anything out because the fraction on the left has two terms in the denominator, and the fraction on the right has two terms in the numerator (see Chapter
21
for info on dividing algebraic terms). However, an important piece of information that you have is that the fraction equals the fraction. So if you cross-multiply these two fractions, you get two results that are also equal:

• x
  (4
  x
  ) = (2
  x
  + 3)(2
  x
  – 2)
  

At this point, you have something you know how to work with. The left side is easy:

The right side requires a bit of FOILing (flip to Chapter
21
for details):

Now all the parentheses are gone, so you can isolate the
x
terms. Because most of these terms are already on the right side of the equation, isolate them on that side:

Combining like terms gives you a pleasant surprise:

• 6 = 2
  x
  

The two
x
²
terms cancel each other out. You may be able to eyeball the correct answer, but here's how to finish:

To check your answer, substitute 3 back into the original equation:

So the answer
x
= 3 is correct.

Chapter 23

In This Chapter

Solving algebra word problems in simple steps

Choosing variables

Using charts

Word problems that require algebra are among the toughest problems that students face — and the most common. Teachers just love algebra word problems because they bring together a lot of what you know, such as solving algebra equations (Chapters
21
and
22
) and turning words into numbers (see Chapters
6
,
13
, and
18
). And standardized tests virtually always include these types of problems.

In this chapter, I show you a five-step method for using algebra to solve word problems. Then I give you a bunch of examples that take you through all five steps.

Along the way, I give you some important tips that can make solving word problems easier. First, I show you how to choose a variable that makes your equation as simple as possible. Next, I give you practice organizing information from the problem into a chart. By the end of this chapter, you'll have a solid understanding of how to solve a wide variety of algebra word problems.

Everything from Chapters
21
and
22
comes into play when you use algebra to solve word problems, so if you feel a little shaky on solving algebraic equations, flip back to those chapters for some review.

Throughout this section, I use the following word problem as an example:

• In three days, Alexandra sold a total of 31 tickets to her school play. On Tuesday, she sold twice as many tickets as on Wednesday. And on Thursday, she sold exactly 7 tickets. How many tickets did Alexandra sell on each day, Tuesday through Thursday?
  

Organizing the information in an algebra word problem by using a chart or picture is usually helpful. Here's what I came up with:

• Tuesday: 	Twice as many as on Wednesday
  Wednesday: 	?
  Thursday: 	7
  Total: 	31

At this point, all the information is in the chart, but the answer still may not be jumping out at you. In this section, I outline a step-by-step method that enables you to solve this problem — and much harder ones as well.

Here are the five steps for solving most algebra word problems:

1. Declare a variable.
   
2. Set up the equation.
   
3. Solve the equation.
   
4. Answer the question that the problem asks.
   
5. Check your answer.
   

As you know from Chapter
21
, a variable is a letter that stands for a number. Most of the time, you don't find the variable
x
(or any other variable, for that matter) in a word problem. That omission doesn't mean you don't need algebra to solve the problem. It just means that you're going to have to put
x
into the problem yourself and decide what it stands for.

 
When you
declare a variable,
you say what that variable means in the problem you're solving.

Here are some examples of variable declarations:

• Let
  m
  = the number of dead mice that the cat dragged into the house.
  
• Let
  p
  = the number of times Marianne's husband promised to take out the garbage.
  
• Let
  c
  = the number of complaints Arnold received after he painted his garage door purple.
  

In each case, you take a variable (
m, p,
or
c
) and give it a meaning by attaching it to a number.

Notice that the earlier chart for the sample problem has a big question mark next to
Wednesday.
This question mark stands for
some number,
so you may want to declare a variable that stands for this number. Here's how you do it:

• Let
  w
  = the number of tickets that Alexandra sold on Wednesday.
  

 Whenever possible, choose a variable with the same initial as what the variable stands for. This practice makes remembering what the variable means a lot easier, which will help you later in the problem.

For the rest of the problem, every time you see the variable
w,
keep in mind that it stands for the number of tickets that Alexandra sold on Wednesday.

After you have a variable to work with, you can go through the problem again and find other ways to use this variable. For example, Alexandra sold twice as many tickets on Tuesday as on Wednesday, so she sold 2
w
tickets on Tuesday. Now you have a lot more information to fill in on the chart:

• Tuesday: 	Twice as many as on Wednesday 	2 w
  Wednesday: 	? 	w
  Thursday: 	7 	7
  Total: 	31 	31

You know that the total number of tickets, or the sum of the tickets she sold on Tuesday, Wednesday, and Thursday, is 31. With the chart filled in like that, you're ready to set up an equation to solve the problem:

• 2
  w
  +
  w
  + 7 = 31
  

After you set up an equation, you can use the tricks from Chapter
22
to solve the equation for
w.
Here's the equation one more time:

• 2
  w
  +
  w
  + 7 = 31
  

For starters, remember that 2
w
really means
w
+
w.
So on the left, you know you really have
w
+
w
+
w,
or 3
w;
you can simplify the equation a little bit, as follows:

• 3
  w
  + 7 = 31
  

The goal at this point is to try to get all the terms with
w
on one side of the equation and all the terms without
w
on the other side. So on the left side of the equation, you want to get rid of the 7. The inverse of addition is subtraction, so subtract 7 from both sides:

You now want to isolate
w
on the left side of the equation. To do this, you have to undo the multiplication by 3, so divide both sides by 3:

You may be tempted to think that, after you've solved the equation, you're done. But you still have a bit more work to do. Look back at the problem, and you see that it asks you this question:

• How many tickets did Alexandra sell on each day, Tuesday through Thursday?
  

At this point, you have some information that can help you solve the problem. The problem tells you that Alexandra sold 7 tickets on Thursday. And because
w
= 8, you now know that she sold 8 tickets on Wednesday. And on Tuesday, she sold twice as many on Wednesday, so she sold 16. So Alexandra sold 16 tickets on Tuesday, 8 on Wednesday, and 7 on Thursday.