122 Part 2: Into the Unknown
Using Equations to Find the Missing Number
Variables can stand for numbers that are unknown or numbers that change. When you use a
variable to take the place of a number that changes values, you usually have an expression, the
mathematical equivalent of a phrase.
For example, if you buy hamburgers for $3.50 each, the amount you have to pay will vary
depending on how many burgers you buy. If you use the variable h to stand for the number
of hamburgers you buy, the amount you have to pay would be represented by the expression
$3.50 × h. If you buy 2 burgers, h is 2 and you pay $7. If you buy 10 burgers, h is 10 and you pay
$35. There’s no one “right” value of h. It’s a different number each time you do the problem.
On the other hand, if you know that hamburgers are $3.50 each, and you know that you spent
$24.50, you can write the equation $3.50 × h = $24.50. Now you have a full mathematical sentence,
an equation, and a question: how many hamburgers did you buy? There’s only one value that can
take the place of h and make that sentence true. Finding that value is the process of solving an
equation.DEFINITION
An expression is a mathematical phrase. Expressions may include variables, but they
do not have an equals sign.
An equation is a mathematical sentence, which often contains a variable.There are times when you can figure out what the value of the variable has to be just by using
your arithmetic facts or by a little guessing and testing. You can call that solving by inspection.
That’s fine when it works, but many times it’s too difficult or too time consuming. You need a
better strategy.
When you have an equation, something happened to the variable. Someone multiplied, or added,
or did something (or several somethings) to the variable, and you know the result. When solving
an equation, your job is to undo the arithmetic that has been performed and get the variable
alone, or isolated, on one side of the equation. For example, if you start with the equation 3x – 2
= 25, someone multiplied the variable by 3, then subtracted 2, and got 25. Your job is to undo that
arithmetic and get to a simple “x = the original number.” In this case, x = 9.
Since you are undoing, you do the opposite of what has been done. The equation is like one
of those old-fashioned scales, with a pan hanging on each side. The same amount of weight is
on each pan, so the scale is balancing. To keep the equation balanced, you perform the same
operation on both sides of the equation. Let’s look at each of the basic steps first and then start
combining them.