Beginning Algebra

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Simplifying Algebraic Expressions


We saw in the prealgebra section that we can combine like terms, like 4n + 2n = 6n.
We also used the distributive law to expand numbers; here we distribute the 6:
6 * 17 = 6(10 + 7) = 6 * 10 + 6 * 7 = 60 + 42 = 102.
Also this works with variables; we state the law usually as a(b + c) = ab + ac.

Example:
In a typical algebra problem you're asked to expand something like 3(x + 5).
3(x + 5) = 3(x) + 3(5) = 3x + 15.

Example:
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.
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 .

Multiplying Algebraic Expressions

Beginning Alebra

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)
= 12x2 + 15x – 48x – 60 (remember to change the sign on that last term)
= 12x2 – 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)
= 12x2 – 48x + 15x – 60  =  12x2 – 33x – 60.  Better!
12x2 - 48 x + 15 x - 60

Example:
(n + 3)(n – 3) = n2 – 3n + 3n – 9 = n2 – 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) = a2 - b2.  Also see the basic factoring and more factoring sections.

Basic Factoring
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 expression  8x3 + 20x2.
Look for the common factor:
8x3 = 2 * 2 * 2 * x * x * x   and   10x2 = 2 * 2 * 5 * x * x.
The common part is  2 * 2 * x * x  =  4x2.  Write it as
8x3  +  20x2 = (4x2)(2x)  +  (4x2)(5) = (4x2)(2x + 5).   That's factored!

Another method that might work (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:  12x2 – 9xy + 8x – 6y .
There's no common factor for all terms, but we can split the pairs:
12x2  –   9xy      +      8x    –   6y
=  3x * 4x – 3x * 3y   +   2 * 4x – 2 * 3y
=  3x(4x – 3y)   +   2(4x – 3y)
=  (3x + 2)(4x – 3y) .

Solving Linear Equations

An equation has to have an equals sign, as in  3x + 5 = 11 .  It's two expressions set equal to one another.
A solution to an equation is a number that can be plugged in for the variable to make a true number statement.

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?
We use some principles of equality, such as:
Adding the same thing (number or variable term) to both sides of an equation
Subtracting the same thing (number or variable term) from both sides
Multiplying or Dividing both sides by a non-zero quantity.
These all keep the equation "balanced" like a scale.

Example:
Let's solve our given equation from scratch:
3x  +  5   =   11        try to isolate  x  so we can see what it is
      – 5      – 5       subtract 5 from each side to get constants on the right
3x         =    6        the intermediate 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 x1 ; no x2 or other powers, no variables in the denominator, and no square roots or other funny stuff.

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