Basic Math and Pre-Algebra For Dummies (100 page)

Generally, you have four ways to solve algebraic equations such as the ones I introduce earlier in this chapter. In this section, I introduce them in order of difficulty.

You can solve easy problems just by looking at them. For example:

• 5 +
  x
  = 6
  

When you look at this problem, you can see that
x
= 1. When a problem is this easy and you can see the answer, you don't need to go to any particular trouble to solve it.

When you can't see an answer just by looking at a problem, sometimes rearranging the problem helps to turn it into one that you can solve using a Big Four operation. For example:

• 6
  x
  = 96
  

You can rearrange this problem using inverse operations, as I show you in Chapter
4
, changing multiplication to division:

Now solve the problem by division (long or otherwise) to find that
x
= 16.

You can solve some equations by guessing an answer and then checking to see whether you're right. For example, suppose you want to solve the following equation:

• 3
  x
  + 7 = 19
  

To find out what
x
equals, start by guessing that
x
= 2. Now check to see whether you're right by substituting 2 for
x
in the equation:

• 3(2) + 7 = 13 WRONG! (13 is less than 19.)
  
• 3(5) + 7 = 22 19 WRONG! (22 is greater than 19.)
  
• 3(4) + 7 = 19 RIGHT!
  

With only three guesses, you found that
x
= 4.

When an algebraic equation gets hard enough, you find that looking at it and rearranging it just isn't enough to solve it. For example:

• 11
  x
  – 13 = 9
  x
  + 3
  

You probably can't tell what
x
equals just by looking at this problem. You also can't solve it just by rearranging it, using an inverse operation. And guessing and checking would be very tedious. Here's where algebra comes into play.

Algebra is especially useful because you can follow mathematical rules to find your answer. Throughout the rest of this chapter, I show you how to use the rules of algebra to turn tough problems like this one into problems that you can solve.

As I show you in the preceding section, some problems are too complicated to find out what the variable (usually
x
) equals just by eyeballing it or rearranging it. For these problems, you need a reliable method for getting the right answer. I call this method the
balance scale.

The balance scale allows you to
solve for x
— that is, find the number that
x
stands for — in a step-by-step process that always works. In this section, I show you how to use the balance scale method to solve algebraic equations.

 The equals sign in any equation means that both sides balance. To keep that equals sign, you have to maintain that balance. In other words, whatever you do to one side of an equation, you have to do to the other.

For example, here's a balanced equation:

If you add 1 to one side of the equation, the scale goes out of balance.

But if you add 1 to
both
sides of the equation, the scale stays balanced:

You can add any number to the equation, as long as you do it to both sides. And in math,
any number
means
x:

• 1 + 2 +
  x
  = 3 +
  x
  

Remember that
x
is the same wherever it appears in a single equation or problem.

This idea of changing both sides of an equation equally isn't limited to addition. You can just as easily subtract an
x,
or even multiply or divide by
x,
as long as you do the same to both sides of the equation:

The simple idea of balance is at the heart of algebra, and it enables you to find out what
x
is in many equations. When you solve an algebraic equation, the goal is to
isolate x
— that is, to get
x
alone on one side of the equation and some number on the other side. In algebraic equations of middling difficulty, this is a three-step process:

1. Get all constants (non-
   x
   terms) on one side of the equation.
   
2. Get all
   x
   -terms on the other side of the equation.
   
3. Divide to isolate
   x.
   

For example, take a look at the following problem:

• 11
  x
  – 13 = 9
  x
  + 3
  

As you follow the steps, notice how I keep the equation balanced at each step:

1. Get all the constants on one side of the equation by adding 13 to both sides of the equation:
   

   

   Because you've obeyed the rules of the balance scale, you know that this new equation is also correct. Now the only non-
   x
   term (16) is on the right side of the equation.
   

2. Get all the
   x
   -terms on the other side by subtracting 9
   x
   from both sides of the equation:
   

   

   Again, the balance is preserved, so the new equation is correct.
   

3. Divide by 2 to isolate
   x:
   

   

   To check this answer, you can simply substitute 8 for
   x
   in the original equation:
   

   

   This checks out, so 8 is the correct value of
   x.