dansmath > lessons > beginning algebra
Arithmetic (The basic operations and what order to do them)
Prealgebra (Introduction to symbols and expressions)
Beginning Algebra (Simplifying, solving, and graphing)

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• Simplifying Expressions
• Solving Linear Equations
• Coordinates and Graphing

• 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!

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.

[ beginning algebra | top of page ]

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).

[ beginning algebra | top of page ]

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. The y is always twice the x. General function, at most one y per x.

More graphs and their equations are available in the functions and graphs section.

Well, that's it for now. Check back often for new stuff!
Click below for other topics, or visit the ask dan page!
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