Consider for now only equations with one unknown variable, denoted by some letter, which can be x but can also be different.
An equation is an equality between two mathematical expressions containing the unknown number (maybe in several places), as well as other numbers which are known. To "solve the equation" means to find which value of the unknown number makes the equality hold true. If (as often happens) the equation arises from an engineering or scientific calculation, its solution then tells what to expect from our design, experiment or observation.
A formula, in contrast, is an equality between two mathematical expressions, which involve two or more unknown numbers, and perhaps also other numbers whose values are given.
By itself a formula does not "have a solution." It is not a puzzle with a certain solution, which needs to be found, but expresses a relationship between the unknown numbers. Many different values of the unknown quantities can exist, each of which makes it come true.
However, if we are given the values of all except one of the unknown numbers, what remains is an equation for the remaining unknown, which can then be solved.
All this is best made clear by examples
The income tax form for my state of Maryland tells me that if my earnings during the year were E dollars (corrected for exemptions, corrections and other tax adjustments), and my state income tax for the year is T dollars, then (with * marking multiplication)
T = (E – 3000)*0.0475 + 90
The above is a formula, and it has no single solution. Each value of E gives a different value of T. Once E is given, this becomes a rather simple kind of equation, and T is immediately found.
In what follows we again use algebraic notation, where touching expressions are understood to be multiplied. Then the * symbol is no longer needed, and the formula becomes
T = (E – 3000) 0.0475 + 90
Two things to note.
we now often denote unknown quantities, not by x and y but by letters which contain clues for what they stand for. Such as "E" for "earnings" and "T" for "tax."
all the tricks used in changing equations from one form to another can be used to change the way a formula may be written. Suppose we know someone's tax T and wish to derive the earnings E.
In this case, the same formula gives a different equation. Earlier it was more convenient to have T stand alone on the left. Now, it is more convenient to isolate E:
Subtract 90 from both sides:
T – 90   = (E – 3000) 0.0475
Divide both sides by 0.0475:
(T – 90) / 0.0475 = (E – 3000)
Add 3000 to both sides
(T – 90) / 0.0475 + 3000 = E
For neater appearance, exchange sides:
E = (T – 90)/0. + 3000
Now, for any T you plug in, you get a value of E.
Temperature is usually measured using either the scale introduced in 1714 by Farenheit or the one proposed in 1742 by Celsius.
If the temperature of some object is F degrees on the scale of Farenheit and C degrees on the scale of Celsius, the two numbers are conncected by the formula
C = (F – 32)(5/9)
Given one of the two numbers, we have an equation for the other.
9C = (F – 32)5
Divide both sides by 5:
(9/5)C = F – 32
Add 32 to both sides
(9/5)C + 32 = F
If F is the one given, the above form immediately gives C. If on the other hand C is given, it helps to isolate F. Multiply both sides by 9:
Exchange sides for better appearance:
F = (9/5)C + 32
The distance s which a
dropped object covers in a time of t seconds, starting from rest, is
s = (1/2) gt2
(algebraic notation, so touching symbols get multiplied).
Here g is the number giving the strength of the Earth's gravity pull: if s is in meters, g = 9.81, if in feet, g = 32.16 (9.81 meters = 32.16 feet). That value is known, but manipulating the formula (as done below) becomes simpler if you keep representing it with a letter until the moment it is actually used.
However, the time t is not given. Whenever you choose a value for t, the formula will give you the appropriate distances.
Suppose we seek the inverse
relationship--given s, what is t? One now views t as the unknown and
proceeds to isolate it. Multiply both sides by 2
2s = gt2
and divide by g
2s/g = t2
To go from t2 to t one must find the square root, an easy task for anyone with a calculator having a square root button (slower methods also exist, using pencil and paper). Mathematics uses a sign √ for this, so
√(2s/g) = t
Now, whatever the distance s is, one can put it in the equation and derive the appropriate time t, in seconds.
Substitution of Formulas
As noted in the first section--discussion of three basic rules of algebra--when the same is done to both sides of an equation, the results are still equal.
The same also holds true for formulas. For instance--if you multiply two sides of a formula by the same number, the result is still a valid formula, even if what you multiply contains unknown numbers. Equal means equal!
The example below is of a sort which often arises. In one of the problems in "Stargazers", one arrives at a formula
VT = 2 p R
where p=3.1415926... is a fixed number, the number of diameters in the length of a circle, T is some time interval and R some distance. Suppose we are told that at instant (1) their values are T1 and R1, and at instant (2), these are T2 and R2. We now have two formulas
VT1 = 2 p R1 (1)
VT2 = 2 p R2 (2)
Any connection between the two pairs, (T1, R1) and (T2, R2)?
Yes! We can divide
After all--we are dividing the sides of (1) by two expressions which are equal, so the results should still be equal, too. We get:
- the left side of (1) by the left side of (2),
- the right side of (1) by the right side of (2)
VT1 / VT2 = 2 p R1 / [2 p R2]
Canceling terms which are the same in numerator and denominator --namely, V on the left and 2 p on the right--leaves
T1 / T2 = R1/ R2
which turns out to be useful in the rest of the calculation. This is a general rule: given two formulas or equations, we may divide each side of one by one of the sides of the other. "Dividing equals by equals, gives results which remain equal".