Algebra Know-It-ALL

In this equation, c does not stand for the speed of light, and e doesn’t mean the exponential
constant. All five constants are meant to stand for ordinary numbers. Letters are used to avoid
specifying exactly what numbers they are. We can, however, place restrictions on them, such
asa> 1, b≥−5,d≠ 0, or c<e.
Letter constants are convenient when we want to show an equation in a certain form. So,
for example, we can write

a+x=b

to represent an equation in x,as long as we realize that a and b represent constants, and as
long as we know that x is the variable. Once we know that the values of a and b always stay the
same no matter what happens to x, we can morph the above equation to solve for x in terms
ofa and b. In this case, it’s easy; we can subtract a from each side to get

x=b−a

Letter constants usually come from the first half of the English alphabet, and are usually writ-
ten in lowercase. Greek letters can also represent constants.

Constants plus or minus x

Here are some first-degree equations that contain a variable x with constants added and/or
subtracted.

x− 4 = 0

x+ 7 =− 2

a−x= 0

a− 5 +x= 0

a−x=b

These equations can be morphed to get x all by itself on the left sides of the equality symbols,
and nothing but constants on the right sides. We can use the rules from Chap. 9 to do this.
The above equations then become:

x= 4

x=− 9

x=a

x= 5 −a

x=a−b

These are the solutions to the original equations, because they clearly state the values of x in
terms of the constants.

Constants, Sums, and Differences 193