Some Simple Equations
Here are some simple equations containing only numbers. Note that these are true no matter what.
3 = 3
3 + 5 = 4 + 4
1,000,000 = 10 ^{6}
− (−20) = 20
Once in a while you’ll see equations containing more than one equals sign and three or more parts. Examples are
3 + 5 = 4 + 4 = 10 − 2
1,000,000 = 1,000 × 1,000 = 10 ^{3} × 10 ^{3} = 10 ^{6}
−(−20) = −1 × (−20) = 20
All the foregoing equations are obviously true; you can check them easily enough. Some equations, however, contain variables as well as numbers. These equations are true only when the variables have certain values; sometimes such equations can never be true no matter what values the variables attain. Here are some equations that contain variables:
x + 5 = 8
x = 2 y + 3
x + y + z = 0
x ^{4} = y ^{5}
y = 3 x − 5
x ^{2} + 2 x + 1 = 0
Variables usually are represented by italicized lowercase letters from near the end of the alphabet.
Constants can be mistaken for variables unless there is supporting text indicating what the symbol stands for and specifying the units involved. Letters from the first half of the alphabet often represent constants. A common example is c , which stands for the speed of light in free space (approximately 299,792 if expressed in kilometers per second and 299,792,000 if expressed in meters per second). Another example is e , the exponential constant, whose value is approximately 2.71828.
Some Simple Formulas
In formulas, we almost always place the quantity to be determined all by itself, as a variable, on the lefthand side of an equals sign and some mathematical expression on the righthand side. When denoting a formula, it is important that every constant and variable be defined so that the reader knows what the formula is used for and what all the quantities represent.
One of the simplest and most wellknown formulas is the formula for finding the area of a rectangle (Fig. 11). Let b represent the length (in meters) of the base of a rectangle, and let h represent the height (in meters) measured perpendicular to the base. Then the area A (in square meters) of the rectangle is
A = bh
A similar formula lets us calculate the area of a triangle (Fig. 12). Let b represent the length (in meters) of the base of a triangle, and let h represent the height (in meters) measured perpendicular to the base. Then the area A (in square meters) of the triangle is
A = bh /2
Consider another formula involving distance traveled as a function of time and speed. Suppose that a car travels at a constant speed s (in meters per second) down a straight highway (Fig. 13). Let t be a specified length of time (in seconds). Then the distance d (in meters) that the car travels in that length of time is given by
d = st
If you’re astute, you will notice something that all three of the preceding formulas have in common: All the units “agree” with each other. Distances are always given in meters, time is given in seconds, and speed is given in meters per second. The preceding formulas for area will not work as shown if A is expressed in square inches and d is expressed in feet. However, the formulas can be converted so that they are valid for those units. This involves the insertion of constants known as conversion factors .

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