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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[ top of page ] Click & choose a topic or just scroll & learn!- Simplifying Expressions
- Solving Linear Equations
- Coordinates and Graphing
Simplifying Expressions[ top of page ] . . . 9/97When 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 ] . . . 9/97An

equationhas to have an equals sign, as in 3x + 5 = 11 .A

solutionto 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
linearis 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 8/01A 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.

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