Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 10: Solving Equations and Inequalities 129

If there are variable terms on both sides of the equation, add or subtract to eliminate one of them.

Next, add or subtract to eliminate the constant term that is on the same side as the variable term.

You want to have one variable term equal to one constant term. Finally, divide both sides by the

coefficient of the variable term.

CHECK POINT

Solve each equation.

11.^11 xx18 3^14 
12.^5240 x 
13.^5476 xx 
14. 45 3xxx ^62 
15. 8 xx 4 16 10  7

Special Cases

Sometimes when you try to isolate the variable, all the variable terms disappear. There are two

reasons why this can happen. Sometimes you’re working with an equation that will make a true

statement no matter what value you substitute for the variable. The simplest example of this is

the equation x = x. No matter what you replace x with, you’ll get a true statement. Equations like

this are called identities. If you subtract x from both sides of the equation, you find yourself with

0 = 0, which is true, but not the “x = a number” you were hoping for. In an identity, x can equal

any number. If all the variables disappear and what’s left is true, you have an identity.

DEFINITION

An identity is an equation that is true for all real numbers.

The other reason why the variables may disappear is that you’re working with an equation that is

never true. The simplest example of this kind of equation is x = x + 1. There’s no way a number

could be equal to more than itself. In this case, if you subtract x from both sides you get 0 = 1,

which is clearly not true. If all the variables disappear and what’s left is false, the equation has no

solution.

Don’t confuse identities or equations with no solution with equations that have a solution of zero.

The equation 0x = 0 is an identity, the equation 0x = 4 has no solution, but the equation 4x = 0

has a solution of x = 0.