Download Chapter 7 Notes

Coursename: MATH120
Course ID: WEISS16313
Chapter 7 - Solving Linear Equations
I. Definitions
A) Algebraic Expression – A group of number or variables connected by +, -, *, / (Except by
Zero), Raising to power or roots.
B) Equation – A statement where two Algebraic Expressions are Equal.
C) Theorem – Linear Equations in one variable: An equation is Linear if and only if it can be
written in the form Ax + B = C where A, B, C are real numbers with A does not equal 0.
D) Solution Set: Group of all possible numbers that work for x in an equation.
E) Equivalent Equations: Equations with the same solution set.
Example:4x +7 = 19
4x = 12
F) Addition Property of Equality – For all Real Numbers A, B, C the equations A = B and A +
C = B +C are equivalent
G) Multiplication Property of Equality – (same as above) and A * C = B * C are equivalent
II. Solving Linear Equations
A) Example: Using Prop. of Equality to solve.
9x – 6x + 15 = x + 9
3x + 15 = x + 9
B) Solving Linear Equations in One Variable.
Step 1: Clear Fractions
Step 2: Simplify each side separately
Step 3: Isolate the variable terms on one side
Step 4: Transform so that coefficient of thevariable is 1
Step 5: Check
III. Conditional Equations, Identities, + Contradictions.
Type
# of Solutions
Indication
Graph
Conditional
1
x=a
Separate
Identity
Infinity
Final Line True 0 = 0
Separate
Contradictions
0
Final Line False 0 =/ 1
IV. Inequalities
A) Addition Property
A) 9 + 10 >
3 + 10
19
>
13
B) Multiplication Property
A) (5)(9) >
6(5)
45
>
30
C) Set Notation {t | t >= 10}
D) Interval Notation [10, ↀ)
E) Graph Notation
-----------------------------------------*--------------->
0
10
V. Quadratic Equations
1) Three Methods for solving quadratics
a) Use zero factor property
Separate
Coursename: MATH120
Course ID: WEISS16313
If a * b = 0, then a = 0 and/or b = 0
Examples:
(x + 4)(x – 6) = 0
m2 + 4m – 5 = 0
(m – 1)(m + 5) = 0
Make m = 1 or -5 to make it equal
15r2 + 7r – 2 = 0
(3r + 2)(5r – 1) = 0
b) Three Methods For Solving Quadratics
1. Square Root Property
If R >=, then the solutions of x2 = R are x = +/- SQ(R) of x2 = R and x = +/- SQ(R)
2) Quadratic Formula
x = -b +/- SQ(b2 – 4ac)
2a
VI. Applications of Linear Regressions
1. Translating words to equations
Words:
{+ Addition, sum, plus, more than, increased by, added to
{- Subtraction, Less than, decrease by, minus, difference, subtracted from
{* Multiply, Times, Of, Product, Twice
{/ Divided by, Quotient
2. General Steps Pg. 356
A) Read
B) Assign a variable
C) Write an equation
D) Solve
E) State the answer
F) Check
3. Types of Problem
A) Find an unknown quantity
Ex.) Pg. 356
Step 2) Let s = # of strike outs
M + J = 573
Step 3) S + (S +95) = 573
25 + 95 = 573
25 = 478
2
2
S = 239
B) Mixture and Interest Problems
Pg. 366 #34 and 43 as examples
C) Motion Problems
D = RT (Distance = Rate * Time)
VII. Variation
1. Direct Variation
Definition – Y varies directly as x
y = kx (k is the variation. This can be in many forms ex. y = kx2 )
Coursename: MATH120
Course ID: WEISS16313
2. Inverse Variation
Definition – Y varies inversely as x
y = k/x (k is the variation again.)
VIII. Exponent and Set Notation
Definition – An = A * A * A * ................. times the number n
Exponent Rules:
1.
Am * An = Am+n
2. A0 = 1
3.
A-n = 1/An
4.
Am/An = Am-n
5.
(Am)n = Am*n
6.
(A * B)m = Am * Bm
7. (A/B)m = Am/Bm