# Math Review Lecture 2

**Lecture 2: Algebra Basics **

**Algebra Basics
• Terminology
**◦ A term is a numerical constant or the product (or

quotient) of a numerical constant and one or more

variables. Examples of terms are 2x and 3x

◦ An algebraic expression is a combination of one or

more terms. Terms in an expression are separated by

either + or – signs.

◦ In the term 3xy, the numerical constant 3 is called a

coefficient. In a simple term such as z, 1 is the

coefficient. A number without any variables is called a

constant term. An expression with one term, such as

3xy, is called a monomial; one with two terms , such as

4a + 2d, is a binomial; one with three terms, such as

xy + z – a, is a trinomial . The general name for

expressions with more than one term is polynomial.

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**• Operations with Polynomials
**◦ All of the laws of arithmetic operations, such as the

associative, distributive, and commutative laws, are also

applicable to polynomials.

◦ All of the laws of arithmetic operations, such as the

associative, distributive, and commutative laws, are also

applicable to polynomials.

◦ **Commutative law**: a + b = b + a

◦ **Associative law: **(x + y) + z = x + (y + z)

◦ **Distributive law:** 2 (a +5) = 2a + 2(5) = 2a + 10

•** Translating English Into Algebra:** In some

word problems, especially those involving

variables, the best approach is to translate

directly from an English sentence into an

algebraic “sentence,” i.e., into an equation. The

table below lists some common English words

and phrases, and the corresponding algebraic

symbols .

**Example:** Steve is now five times as old as Craig was 5 years

ago. If the sum of Craig’s and Steve’s ages is 35, in how many

years will Steve be twice as old as Craig?

**Lecture 2: Orders of Operation**

**Order of Operations
• PEMDAS:** “Please Excuse My Dear Aunt Sally.”

Parentheses, exponents, multiplication, division,

addition, subtraction.

◦ Start with the innermost set of parentheses and work

outwards.

◦ Absolute value indicators are treated as parentheses.

◦ Multiplication & division are interchangeable

◦ Addition & subtraction are interchangeable

**Lecture 2: Equations**

**Equations
**•

**Equations**: An equation is an algebraic sentence

that says that two expressions are equal to each

other. The two expressions consist of numbers,

variables, and arithmetic operations to be

performed on these numbers and variables.

**• A linear or first -degree equation is an
**equation in which all the variables are raised to

the first power. (There are no squares or cubes .)

n + 6 = 10

**• A quadratic or second-degree equation
**contains a squared term and no greater power.

The equation can be written as:

ax^{2} + bx + c = 0

where a is not equal to zero .

**Lecture 2: Variation**

**Variation
**•

**Direct Variation:**One variable increases when

the other increases, and decreases when the

other decreases. For instance, the amount of

paint needed to paint a wall varies directly with

the area of the wall. When two quantities x and y

vary directly, their relationship can be expressed

by the equation y = kx, where k is a constant.

**• Inverse Variation:** In inverse variation, one

variable decreases when the other increases, and

increases when the other decreases. When two

quantities x and y vary inversely, their

relationship can be expressed by xy = k, or y =

k/x. One inverse relationship is rate × time =

distance. To cover a constant distance in less

time (decreasing time), you must go faster

(increasing rate).

**Lecture 2: Functions**

**Functions
• Definition of a function:
**A function is a rule

that assigns each element in the domain to one

and only one element in the range.

• The domain is the set of all possible numbers

that can be used as an input. This is also called

the explanatory or independent variable.

• The range is the set of all possible values that

are the output. This is also known as the

response variable or dependent variable.

•** Functional Notation:** y = f(x) = mx + b

• The domain is x and the range is f(x)

• For public policy applications, we must consider

the practical as well as mathematical meaning.

For example, it might not make sense to have a

negative quantity of books or a negative price.

**Lecture 2: Correlation vs. Causation**

**• Correlation:** Variables vary together, either

directly or inversely.

**o Examples:
**o A person’s height and weight

o Extent of fire damage and number of firefighters

**• Causation**: an event B always occurs when A

occurs (deterministic); the occurrence of A

increases the probability of B (probabilistic)

**o Examples:
**o Rain and a wet roof

o Air pollution and respiratory illness

**Lecture 2: Solving Equations**

**Solving Equations
**• The goal is to isolate the variable you’re solving

for on one side.

• What you do to one side, e.g. subtracting x, you

must always do to the other side.

• Add terms together, but only if they have the

same exponent. This process is known as

collecting like terms .

• Follow the order of operations.

**• Adding and Subtracting Monomials: **To

combine terms with the same variable and

exponent, keep the variable part unchanged

while adding or subtracting the coefficients.

**o Examples:
**2a + 3a =

10x – 2x =

**• Adding and Subtracting Polynomials:** Again,

combine like terms, where the exponent is equal.

**o Example:**

(3x^{2} + 5x – 7) – x^{2} =

•** Remove parenthesis ** by distributing the

coefficient, as in 3(2y – 4) = 6y – 12

•** Divide both sides by the coefficient **to get

the variable by itself.

• Eliminate fractions by multiplying both sides

by the lowest common denominator . Or, if two

fractions are equal to each other, cross

multiply. This is a shortcut in which you simply

go from to

• Example of an equation with fractional

coefficients:

• Solve by multiplying both sides by the LCD ,

which is 30. Then distribute and rearrange.

**• Substitute** known quantities in an expression

◦ Example: If x=2, replace x with 2 wherever x appears

**◦ Example:** Suppose a person’s salary (S) is influenced

both by her years of education (E) and her

parents’ income(I):

◦ Jason has a high school diploma; his parents’ average

salary was $80,000. Rachel has an MPA; her parents

earned $55,000. Who has the greater earning potential?

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