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- Arithmetic (The basic operations and what order to do them)
- Prealgebra (Introduction to symbols and expressions)
- Beginning Algebra (Simplifying, solving, and graphing)
- (c)
1997-2002 Dan Bach and B & L Math Enterprises; all rights
reserved. Download for personal use only.
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& choose a topic or just scroll & learn!
Simplifying Expressions [
top of page ] . . . 9/97
When faced with an expression
like 4x + 5(3x - 12), what do we do first? Let's see: PEMDAS
says work in parentheses first, but 3x and 12 are unlike. Hmm,
let's try the distributive
law:
- 4x + 5(3x - 12)
- = 4x + 5(3x) - 5(12)
- = 4x + 15x - 60
- = 19x - 60 . This problem
was no problem!
What
about (4x + 5)(3x - 12) ? Is this the same as 4x + 5(3x - 12)
?
No, the parentheses change
it. Here we can use the distributive law twice:
(4x + 5)(3x - 12)
= (4x + 5)(3x) - (4x
+ 5)(12)
= 12x^2 + 15x - 48x - 60 (remember
to change the sign on that last term)
=
12x^2 - 33x - 60 . That worked, but it was long.
Is this the only way? No.
The best way? No. Use the "FOIL system":
First, Outside, Inside,
Last.
- (4x + 5)(3x - 12)
- . . . . F . . . . . . O .
. . . . . I . . . . . . L . . . .
- = (4x)(3x) - (4x)(12) + (5)(3x)
- (5)(12)
- = 12x^2 - 48x + 15x - 60 = 12x^2 -
33x - 60. Better!
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- Here's another example: . . .
- (n + 3)(n - 3) = n^2
- 3n + 3n - 9 = n^2 - 9.
- Notice the "middle terms"
cancel, and we're left with what's called the difference of
two squares.
- In general, (a + b)(a - b)
= a^2 - b^2. Also
see the factoring
section.
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- [ beginning algebra |
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Solving Linear Equations [
top of page ] . . . 9/97
An equation
has to have an equals sign, as in 3x + 5 = 11 .
A solution
to an equation is a number that can be plugged in for the variable
to make a true number statement.
For example, putting 2
in for x above in 3x + 5 = 11 gives
3(2) + 5 = 11
, which says 6 + 5 = 11 ; that's true! So 2 is a solution.
But how to start with the
equation, and get (not guess) the solution?
3x + 5 = 11 . . . our
given equation
- 5 . . . . . . . - 5 . . . .subtract 5 from
each side to get constants on the right
3x = 6 . . . . . . . . . . the
result
3x / 3 = 6 / 3 . . divide both
sides by 3 to isolate the x
x = 2 . . . . . . . . . . . the
solution (same as before!)
. . . . . . . . . . . . . . . . . We've solved the equation.
- The thing that makes this
equation linear is that the highest power of x is
x^1
- (no x^2 or other powers;
for those "quadratic equations" go to intermediate
algebra).
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- [ beginning algebra |
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Coordinates and Graphing [
top of page ] . . . 9/97, revised 8/01
A point on the screen you're
looking at (like this red one: .) has a "location" which
is measured by how many pixels across and down it is from the
upper left corner. These are its "screen coordinates."
In math, the coordinates of
a point in the plane are measured in relation to a "central"
point, the origin, first to the right, then up.
The coords are listed as (x,
y) for (over, up). In the picture, the origin is at the + and the red dot has coords (x, y) = (5, 2).
. ^y
..4|......
..3|......
..2|......
..1|......
--0+------>x
.-10123456
.-2|......
- Coordinates are also used
in writing equations for graphs; we can have a relation between
- x and y, and translate that
into the language of pictures.
- In the first two examples,
the functions are "linear" so the graphs are straight
lines.
The x and y coords add up to
2.
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The y is always twice the x.
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General function, at most one
y per x.
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- More graphs and their equations
are available in the functions
and graphs section.
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