# Note Sheet #3

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"Note Sheet #3"

```					Note Sheet #3        Solving Linear Systems Algebraically
Rules of exponents and Integer exponents

Vocabulary
Consistent – System has a solution (the line intersect)
Inconsistent – System does not have a solution (ie parallel lines)
Dependent - infinite number of solutions (the equations graph the same line!)

Algebraic Methods
The Algebraic Methods are designed to find the coordinates of the points of
intersection that you would find if you graphed the two lines (the solution to the
system of equations). Both algebraic methods find ONLY one half of the
solution. To find the "partner" in the ordered pair you will need to evaluate one of
the equations with using the value you found. You should check your work by
evaluating the other equation using the ordered pair to verify your work.

Substitution Method: You must solve one of the equations for one of the
variables. (This is the preferred method is used when one of the coefficients is a
ONE to avoid fractions or is already solved for one of the variables). Then you
"substitute" the information from that equation into the second equation. Example:

x = 3y - 1   (EQ. 1)
2x + 5y = 7       (EQ. 2)
2(3y - 1) + 5y = 7 (Combo of EQ. 1 & 2)

Elimination Method:          Both equations must be written in standard form.
( Ax + By = C) The object is to add to two equations together and in the process
one of the variables (letters) is eliminated. Most of the time we have to multiply
everything on both sides of the equal sign by a number so that they will "cancel out"
Example 1: 3x + 2y = 5         Example 2:        -3x + 2y = 5 (EQ. 1)
5x - 2y = 11                         4x - 3y = -9 (EQ. 2)
8x     = 16
x=2                               3(-3x + 2y) = (5)3 (EQ 1)
2(4x - 3y) = (-9)2 (EQ 2)

-9x + 6y = 15      (EQ. 1)
8x - 6y = -18     (EQ. 2)

x    = -3 (Combo)
Important
Neither Algebraic Method you the solution. It gives you ½ of the solution. You
need to take the coordinate and find the other ½ of the ordered pair.by “plugging it
in” to one of the original equations. Always check your final solution in in the
“other equation”.
Exponential Notation - a "shorthand" way of writing a multiplication problem.
Exponent only effects the number its next too. If it is to effect more than just the
number it has to be "bundled together" (bundle)exponent

Definition Exponents

Positive exponent                 negative exponent              zero exponents
1
x4 = (x)(x)(x)(x)                        x −3 = 3                x0 = 1, x ≠ 0
x

It is helplful to recognize the pattern of a negative exponent in the denominator of a
fraction.
2 2 x3                  3x −2 3 y 3      3x 2 y −1 3x 2 w3
=     = 2 x3              = 2                 =
x −3   1                 5 y −3 5 x        w−3 z 5   yz 5

Product, Quotient and Power “rules” are just recognizing and using patters that
occur when we apply the definitions of exponents.

x 5 x 7 = x12   Product Rule

x8              x5
x5
= x3
x 8
= 3
x
1
( aka   x −3 ) Quotient Rule

(x )
2 4
= x8    Power Rule