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)
- (c) 1997-2007 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.

[ top of page ] Click & choose a topic or just scroll & learn!- Simplifying Expressions
- Basic Factoring Techniques
- Solving Linear Equations
- Solving Word Problems
- Coordinates and Graphing
Simplifying Expressions[ top of page ] . . . 9/97 - revised 10/07We saw in the prealgebra section that we can combine like terms, like 2n + 3n = 5n.

We also used the distributive law to expand numbers; here we

distribute the 7:

7*13 =7(10 + 3) =7*10 +7*3 = 70 + 21 = 91.Also this works with variables; we state the law usually as

a(b + c) = ab + ac.In a typical algebra problem you're asked to expand something like

5(x + 7).

5(x + 7) =5(x) +5(7) = 5x + 35. This problem was. . . no problem!

- 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 again, but just with the multiplied 5:

- 4x +
5(3x - 12)- = 4x +
5(3x) -5(12)- = 4x + 15x - 60
- = 19x - 60 . That's our answer, simplified.

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 basic factoring and more factoring sections.[ beginning algebra | top of page ]

Basic Factoring[ more factoring | top of page ] . . . 10/07

- We saw the distributive law work to expand things out:

- 6(2x + 7) = 6 * 2x + 6 * 7 = 12x + 42.
- The steps can be reversed to give a
factorization:

12x + 42=6* 2x +6* 7 =6(2x + 7).- This is called factoring out a
common factor.In this case the common factor was 6.Example: Factor the expression8 x^3 + 20 x^2.- Look for the common factor:

- 8 x^3 =
2*2*2*x*x*x and 10x^2 =2*2*5*x*x.- The common part is
2*2*x*x = 4 x^2.Write it as

- 8 x^3 + 20 x^2 = (4 x^2)(2x) + (4 x^2)(5) =
(4 x^2)(2x + 5).- That's factored!
- Another method that works if we're lucky is called the
grouping method.This works by- grouping up equal size blocks of terms and factoring out a common factor from each, then
- hoping the "left-over factors" are equal. That's where the luck, or sometimes skill, comes in.
Example:Factor by grouping:12x^2-9xy + 8x-6y.- There's no common factor for all terms, but we can split the pairs:
- 12x^2 - 9xy + 8x - 6y = 3x
(4x- 3y)+ 2(4x- 3y)= (3x + 2)(4x - 3y) .- Click here for more factoring in the Intermediate Algebra section.
[ beginning algebra | top of page ] Solving Linear Equations[ top of page ] . . . 9/97 . . . revised 11/07

- An
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
2is a solution.- But how to start with the equation, and get (not guess) the solution?
- We use some
principles of equality,such as:

- *
Addingthe same thing (number or variable term) toboth sidesof an equation- *
Subtractingthe same thing (number or variable term) from both sides- *
Multiplyingordividingboth sides by a non-zero quantity.- These all keep the equation "balanced" like a scale.
3x + 5 = 11. . . our given equation- . . . - 5 . . . - 5 . . . .subtract 5 from each side to get constants on the right
3x. . .= 6. . . . . . . . . . the intermediate result3x/ 3= 6/ 3 . . divide both sides by 3 to isolate the x- . . . .
x = 2. . . . . . . the solution (same as before!) . . . . . . We'vesolved the equation.

- The thing that makes this equation
linearis that the highest power of x is x^1 ; no- x^2 or other powers, variables in the denominator, square roots, or other funny stuff.
- (For "quadratic equations" go to intermediate algebra).

[ beginning algebra | top of page ] Solving Word Problems[ top of page ] . . . 11/07

- A
word problemis a mathematical equation phrased in regular language.- Students groan when we tackle word problems, but they also ask,
- "What are we gonna use this stuff for?" The answer is, word problems!
- There's no one rule for solving all word problems, but you'll increase your
- chance of success if you follow a good
strategy. Try thisseries of steps:

STRATEGY TO TRANSLATE AND SOLVE WORD PROBLEMS

Step 1.Readthe problem. Figure out what quantities are known, and what are unknown.Step 2.Decide "What am I being asked to find?" Write thisunknowndown and thengive it a letter name. Put other quantities in terms of this variable.Step 3. Translatethe sentences into mathematical symbols. Look for key words, andform the equationthat relates what you want to what you know.Step 4.Now use algebra techniques tosolvethe equation for the unknown quantity.Step 5. Restateyour answer in English, using Step 2, to give a clear answer to themain question that was asked. Step 6.Checkyour value for the unknown in the original words of the problem.Now you're done!

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.

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