Algebra Demystified 2nd Ed

184 algebra De mystif ieD

algebra DeMYSTiFieD / gibilisco / 000-0 / Chapter 4

In a linear equation, the variables are raised to the first power—there are no

variables in denominators, no variables to any power (other than one), and no

variables under root signs. A linear equations contains an unknown, usually only

one but possibly several. What is meant by the phrase, “solve for x” is to isolate

x on one side of the equation and to move everything else on the other side.

Usually, although not always, the last line is the sentence: “x = (number),” where

the number satisfies the original equation. That is, when the number is substi-

tuted for x, the equation is true.

In the equation 3x + 7 = 1; x = –2 is the solution because 3(–2) + 7 = 1 is a

true statement. For any other number, the statement would be false. For

instance, if we were to say that x = 4, the sentence would be 3(4) + 7 = 1, which

is false. Not every equation has a solution. For example, x + 3 = x + 10 has no

solution. Why not? There is no number that can be added to 3 and be the same

quantity as when it is added to 10. If you were to try to solve for x, you would

end up with the false statement 3 = 10.

In order to solve equations and to verify solutions, we must follow the order

of operations. For example, in the formula:

s xy zy

n

= −+−

−

()^22 ()

1

what operation is done first? Second? Third? A mnemonic for remembering

operation order is “Please excuse my dear Aunt Sally.”

P—parentheses first

E—exponents (and roots) second

M—multiplication third

D—division third (multiplication and division should be done together,

working from left to right)

A—addition fourth

S—subtraction fourth (addition and subtraction should be done together,

working from left to right)

When working with fractions, think of numerators and denominators as being

in parentheses.