Page 51 - MarceAlgebra Demystified

38 CHAPTER 2 Introduction to Variables

3ð23Þþ 4 ¼ 73 and 7ð23Þ¼ 161, so we can conclude that in general
3x þ 4 6¼ 7x. (Actually for x ¼ 1, and only x ¼ 1, they are equal.)
This method for checking equality of algebraic expressions is not
foolproof. Equal numbers do not always guarantee that the expressions
are equal. Also be careful not to make an arithmetic error. The two expres-
sions might be equal but making an arithmetic error might lead you to
conclude that they are not equal.

Canceling with Variables

Variables can be canceled in fractions just as whole numbers can be.

Examples

2x 2 x
¼ ¼ 2
x 1 x
6x 6 x 6 2
¼ ¼ ¼
9x 9 x 9 3
7xy 7y x 7y
¼ ¼
5x 5 x 5
When you see a plus or minus sign in a fraction, be very careful when you
2 þ x
cancel. For example in the expression , x cannot be canceled. The only
x
quantities that can be canceled are factors. Many students mistakenly ‘‘can-
2 þ x 2 þ 1 2 þ x
cel’’ the x and conclude that ¼ ¼ 3or ¼ 2. These equations
2 þ x x 1 x
are false. If were equal to 2 or to 3, then we could substitute any value
x
for x (except for 0) and we would get a true equation. Let’s try x = 19:
2 þ 19 21
¼ .
19 19
2 þ x 2 þ x
We can see that 6¼ 2 and 6¼ 3. The reason that the x cannot be
x x
factored is that x is a term in this expression, not a factor. (A term is a
quantity separated from others by a plus or minus sign.) If you must cancel
2 þ x
the x out of , you must rewrite the fraction:
x