Algebra is a way of doing math with letters
instead of just numbers. The numbers are called variables
if their values can change, and constant if the values
are always the same. Instead of
6 = 2 X 3
we can say y=ax (read "y equals a times x") if
we say y=6, a=2, x=3.
Numbers and letters can be mixed. If we say
y = 0.1x
we say the same as y=ax above, where a is set to
0.1, a constant value.
We can solve an algebraic equation for any one
variable, provided we know all the others. If y= 0.1x
above, we can say y=1 if and only if x=10.
We can rearrange any equation in order to solve for
any of the variables. The equations
y = ax y/x = a
x = y/a
are equivalent. Proof:
- y = ax multiply both sides by 1/x
- (1/x) X y = ax X (1/x) becomes y/x = ax/x
= a(x/x)
- x/x = 1, so a = y/x
We could also multiply both sides by 1/a to get x =
y/a.
Problem: Prove that the statements below are
equivalent
- y = 3x + b
- b = y - 3x
- x = (y - b) / 3
Now verify that they are equivalent by plugging
in the numbers
y = 10, b = 4, x = 2.
Answer
Problem: Express F = GmM / ( r2 ) in
four other ways.
Answer
Besides the four basic mathematical functions,
there are others:
Inversion
Basically, this is taking "one over" a number or
variable. The inverse of 2 is 1/2, or one-half. The
inverse of x is 1/x ("one over x"). If you invert one
side of an equation, you must invert the other side!
If y = ax, then 1 / y = 1 / (ax)
Inverting a fraction is a simple matter of
"flipping it over."
If y = a / b, then 1 / y = b / a
Problem:
- If 1 / d = x / y then d =
- If 1 / d = (x + b) / y then d =
- If 1 / d = b + (x / y) then d =
( Hint: b = (by) / y )
Answer
Exponentiation
The most common example of this process is squaring:
x2 = x times x. But there are others:
x3 = x times x times x
q6 = q X q X q X q X q X q
Since in normal algebra this would be
x3 = xxx and q6 = qqqqqq, you can
hopefully see why exponents were invented!
Roots
Roots are the opposite of exponents, and "undo"
exponents:
x = square root(x2)
x = cube root(x3)
Roots are often expressed as "inverse
exponents." The square root of x is x1/2, the
cube root of x is x1/3, etc. See the pattern?
Problem:
If a3 = p2 and a = 2
then p = ?
If p = 8 then a = ?
Answer
Roots and exponents have the following
properties:
- x6 =
(x3) (x3) =
x3+3
- x6 =
(x2) (x2) (x2)
= x2+2+2
- (xy)4 =
x4 y4
- x7 /
x2 = x7-2 =
x5
When two terms with exponents and a common variable
are multiplied, we add the exponents. When such terms are
divided, we subtract the "bottom" exponent from the "top"
one.
Four is the highest root / exponent we will deal with
in this class.
Problems:
- x5 = (x3)(?)
- x3y3 =
(xy)?
- x2 / y2 = (x /
y)?
- x5 / x2 = ?
Answers
Problem: Write the expression for s is f = n /
s2
Solve the expression you derived numerically for f =7
and n=63.
Answer
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Updated
8/16/99
By James
E. Heath