Solving for the Unknown - Basic Mathematics for Water and Wastewater Operators - page 113

8 Solving for

the Unknown

MATH RULES OF OPERATION

1. Work from left to right.

2. Do all of the work inside the parentheses first.

3. Do all of the multiplication and division above the line (numerator).

4. Do all of the addition and subtraction above and below the line.

5. Perform the division (divide the numerator by the denominator).

Many water/wastewater calculations involve the use of formulas and equations; for
example, process control operations may require the use of equations to solve for an
unknown quantity. To make these calculations, you must first know the values for all
but one of the terms of the equation to be used. What is an equation? Put simply, an
equation is a mathematical statement telling us that what is on one side of an equal
(=) sign is equal to what is on the other side; for example, 4 + 5 = 9. Now, suppose we
decide to add 4 to the left side (4 + 4 + 5). What must we do then? We must also add
4 to the right side (4 + 9). Consider the equation 6 + 2 = 8. If we subtract 3 from the
left side (6 + 2 - 3), what must we do next? We must also subtract 3 from the right
side (8 - 3). It follows that if the right side were multiplied by a certain number, we
must also multiply the left side by that same number. Finally, if one side is divided
by, for example, 4, then we must also divide the other side by 4.
The bottom line is that what we do to one side of the equation we must also do to
the other side. This is the case, of course, because the two sides, by definition, are
always equal.

EQUATIONS
An equation is a statement that two expressions or quantities are equal in value. The
statement of equality 6 x + 4 = 19 is an equation; that is, it is algebraic shorthand for
“The sum of 6 times a number plus 4 is equal to 19.” It can be seen that the equation
6 x + 4 = 19 is much easier to work with than the equivalent sentence. When thinking
about equations, it is helpful to consider an equation as being similar to a balance.
The equal sign tells you that two quantities are “in balance” (i.e., they are equal).
Referring back to the equation 6 x + 4 = 19, it can be solved as follows:

1. 6 x + 4 = 19

2. 6 x = 15

3. x = 2.5

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