Solving for the Unknown - Basic Mathematics for Water and Wastewater Operators - page 115

How can we determine the value of x ? By following the axioms presented below,
solving for the unknown is quite simple.
Note: It is important to point out that the following discussion includes only what
the axioms are and how they work.
AXIOMS
1. If equal numbers are added to equal numbers, the sums are equal.
2. If equal numbers are subtracted from equal numbers, the remainders are
equal.
3. If equal numbers are multiplied by equal numbers, the products are equal.
4. If equal numbers are divided by equal numbers (except zero), the quotients
are equal.
5. Numbers that are equal to the same number or to equal numbers are equal
to each other.
6. Like powers of equal numbers are equal.
7. Like roots of equal numbers are equal.
8. The whole of anything equals the sum of all its parts.
Note: Axioms 2 and 4 were used to solve the equation 6 x + 4 = 19.
SOLVING EQUATIONS
Solving an equation requires determining the value or values of the unknown num-
ber or numbers in the equation.
■ Example 8.2
Problem: Find the value of x if x - 8 = 2.
Solution: Here it can be seen by inspection that x = 10, but inspection does not help
in solving more complicated equations. However, notice that to determine that x = 10
we can add 8 to each side of the given equation; thus, we have acquired a method or
procedure that can be applied to similar but more comple x problems.
x - 8 = 2
Add 8 to each member (axiom 1):
x = 2 + 8
Collect the terms (that is, add 2 and 8):
x = 10