Basic Math and Pre-Algebra For Dummies (101 page)

When you understand how algebra works like a balance scale, as I show you in the preceding section, you can begin to solve more-difficult algebraic equations. The basic tactic is always the same: Changing both sides of the equation equally at every step, try to isolate
x
on one side of the equation.

In this section, I show you how to put your skills from Chapter
21
to work solving equations. First, I show you how rearranging the terms in an expression is similar to rearranging them in an algebraic equation. Next, I show you how removing parentheses from an equation can help you solve it. Finally, you discover how cross-multiplication is useful for solving algebraic equations with fractions.

Rearranging terms becomes all-important when working with equations. For example, suppose you're working with this equation:

• 5
  x
  – 4 = 2
  x
  + 2
  

When you think about it, this equation is really two expressions connected with an equals sign. And of course, that's true of
every
equation. That's why everything you find out about expressions in Chapter
21
is useful for solving equations. For example, you can rearrange the terms on one side of an equation. So here's another way to write the same equation:

• – 4 + 5
  x
  = 2
  x
  + 2
  

And here's a third way:

• – 4 + 5
  x
  = 2 + 2
  x
  

This flexibility to rearrange terms comes in handy when you're solving equations.

Earlier in this chapter, I show you how an equation is similar to a balance scale. For example, take a look at Figure 
22-1
.

To keep the scale balanced, if you add or remove anything on one side, you must do the same on the other side. For example:

Now take a look at these two versions of this equation side by side:

• 2
  x
  – 3 = 11   –3 = 11 – 2
  x
  

In the first version, the term 2
x
is on the left side of the equals sign. In the second, the term –2
x
is on the right side. This example illustrates an important rule.

 When you move any term in an expression to the other side of the equals sign, change its sign (from plus to minus or from minus to plus).

As another example, suppose you're working with this equation:

• 4
  x
  – 2 = 3
  x
  + 1
  

You have
x
's on both sides of the equation, so say you want to move the 3
x.
When you move the term 3
x
from the right side to the left side, you have to change its sign from plus to minus (technically, you're subtracting 3
x
from both sides of the equation).

• 4
  x
  – 2 – 3
  x
  = 1
  

After that, you can simplify the expression on the left side of the equation by combining like terms:

• x
  – 2 = 1
  

At this point, you can probably see that
x
= 3 because 3 – 2 = 1. But just to be sure, move the –2 term to the right side and change its sign:

To check this result, substitute a 3 wherever
x
appears in the original equation:

As you can see, moving terms from one side of an equation to the other can be a big help when you're solving equations.

Chapter
21
gives you a treasure trove of tricks for simplifying expressions, and they come in handy when you're solving equations. One key skill from that chapter is removing parentheses from expressions. This tactic is also indispensable when you're solving equations.

For example, suppose you have the following equation:

• 5
  x
  + (6
  x
  – 15) = 30 – (
  x
  – 7) + 8
  

Your mission is to get all the
x
terms on one side of the equation and all the constants on the other. As the equation stands, however,
x
terms and constants are “locked together” inside parentheses. In other words, you can't isolate the
x
terms from the constants. So before you can isolate terms, you need to remove the parentheses from the equation.

Recall that an equation is really just two expressions connected by an equals sign. So you can start working with the expression on the left side. In this expression, the parentheses begin with a plus sign (+), so you can just remove them:

• 5
  x
  +
  6
  x
  – 15
  = 30 – (
  x
  – 7) + 8
  

Now move on to the expression on the right side. This time, the parentheses come right after a minus sign (–). To remove them, change the sign of both terms inside the parentheses:
x
becomes –
x,
and –7 becomes 7:

• 5
  x
  + 6
  x
  – 15 = 30
  –
  x
  + 7
  + 8
  

Bravo! Now you can isolate
x
terms to your heart's content. Move the –
x
from the right side to the left, changing it to
x:

• 5
  x
  + 6
  x
  – 15
  +
  x
  = 30 + 7 + 8
  

Next, move –15 from the left side to the right, changing it to 15:

• 5
  x
  + 6
  x
  +
  x
  = 30 + 7 + 8
  + 15
  

Now combine like terms on both sides of the equation:

Finally, get rid of the coefficient 12 by dividing:

As usual, you can check your answer by substituting 5 into the original equation wherever
x
appears:

Here's one more example:

• 11 + 3(–3
  x
  + 1) = 25 – (7
  x
  – 3) – 12