Chapter 1: Variables, Real Numbers, and Mathematical Models – Study Notes

Variables, Real Numbers, and Mathematical Models

Introduction to Algebra: Variables and Mathematical Models

This chapter introduces the foundational concepts of algebra, focusing on variables, algebraic expressions, equations, and mathematical models. Understanding these concepts is essential for solving real-world problems using algebraic methods.

Variables and Algebraic Expressions

• Variable: A letter that represents a variety of different numbers. Common variables include x, y, and z.

• Algebraic Expression: A combination of variables and numbers using operations such as addition, subtraction, multiplication, division, powers, or roots. Examples: 3x + 5, x^2 - 7.

Example: The expression 5 + 3x is algebraic because it combines a variable and numbers using addition and multiplication.

Order of Operations

To evaluate algebraic expressions correctly, follow the order of operations:

• 1. Perform all operations within grouping symbols (parentheses, brackets).

• 2. Carry out multiplications and divisions from left to right.

• 3. Complete additions and subtractions from left to right.

Example: Evaluate 5 + 3x for x = 2:

• Replace x with 2: 5 + 3(2)

• Multiply: 5 + 6

• Add: 11

Example: Evaluate 5(x + 7) for x = 2:

• Replace x with 2: 5(2 + 7)

• Add inside parentheses: 5(9)

• Multiply: 45

Translating English Phrases into Algebraic Expressions

Algebra often requires translating verbal statements into mathematical expressions.

• "The sum of a number and 7" → x + 7

• "Ten less than a number" → x - 10

• "Twice a number, decreased by 6" → 2x - 6

• "The product of 8 and a number" → 8x

• "Three more than the quotient of a number and 11" → (x / 11) + 3

Equations and Solutions

An equation is a statement that two algebraic expressions are equal, always containing the equality symbol =. A solution to an equation is a value for the variable that makes the equation true.

• Example: Is x = 5 a solution to 6x + 16 = 46?

• Substitute: 6(5) + 16 = 30 + 16 = 46

• Since both sides equal 46, x = 5 is a solution.

• Example: Is z = 7 a solution to 2(z + 1) = 5(z - 2)?

• Substitute: 2(8) = 16, 5(5) = 25

• Since 16 ≠ 25, z = 7 is not a solution.

Translating English Sentences into Algebraic Equations

Translating sentences into equations is a key skill in algebra. Look for words that indicate equality, such as "equals," "gives," "yields," "is the same as," or "is/was/will be."

• "The product of 8 and a number is 48" → 8x = 48

• "Nine less than 4 times a number gives 26" → 4x - 9 = 26

Formulas and Mathematical Models

A formula is an equation that expresses a relationship between two or more variables. The process of finding formulas to describe real-world phenomena is called mathematical modeling. These formulas, along with the meaning assigned to the variables, are called mathematical models.

• Example: The formula for estimating temperature based on cricket chirps is a mathematical model.

Mathematical Models: Probability of Divorce by Wife's Age at Marriage

Mathematical models can be used to approximate real-world data. For example, the probability of divorce can be modeled based on the wife's age at marriage.

• Model 1: Wife under 18 at marriage: $d = 4n + 5$

• Model 2: Wife over 25 at marriage: $d = 2.3n + 1.5$

• Where n is the number of years after marriage and d is the percentage of marriages ending in divorce.

Example: For a wife under 18, after 15 years:

• $d = 4(15) + 5 = 60 + 5 = 65$

• So, 65% of marriages end in divorce after 15 years (model prediction).

According to the line graph, the actual percentage is 60%. The model overestimates by 5%.

Wife's Age at Marriage	Mathematical Model	Predicted Divorce Rate after 15 Years	Actual Divorce Rate (Graph)	Difference
Under 18	$d = 4n + 5$	65%	60%	+5%
Over 25	$d = 2.3n + 1.5$	36%	36%	0%

Additional info: Mathematical models are useful for making predictions, but they may not always match real-world data exactly. Differences can arise due to simplifications or assumptions in the model.