A variable is a symbol for a number whose value is unknown. A variable might represent quantities at different times. For example if you are paid by the hour for your job and you earn $10 per hour, letting x represent the number of hours worked would allow you to write your earnings as ‘‘10x’’. The value of your earnings varies depending on the number of hours worked. If an equation has one variable, we can use algebra to determine what value the variable is representing.
Variables are treated like numbers
because they are numbers. For instance 2 + x means two plus the quantity x
and 2x means two times the quantity x (when no operation sign is given, the
operation is assumed to be multiplication). The expression 3x + 4 means three times x plus four. This is not the same as 3x + 4x which is three x’s
plus four x’s for a total of seven x’s: 3x + 4x = 7x.
If you are working with variables and
want to check whether the expression you have computed is really equal to the expression with
which you started, take
some larger prime number, not a factor of anything else in the expression, and plug it into both the original
expression and the last one. If the resulting numbers are the same, it is very likely that the first
and last expressions are equal.
For example you might ask ‘‘Is it true that 3x + 4 = 7x?’’ Test x = 23: 3(23) + 4 = 73
and 7(23) = 161, so we can conclude that in general 3x + 4 ≠ 7x.
(Actually for x = 1, and only x = 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 expressions might be equal but
making an arithmetic error might lead you to conclude that they are not equal.
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