MAT 204 Module 1: Introduction to Algebra

Overview of Algebra

• Algebra involves symbols and rules for manipulating them, essential for solving equations and understanding relationships between quantities.
• Variables represent unknown quantities, allowing abstract problem-solving beyond specific numbers.

Classification of Numbers

• Real Numbers: Commonly used in everyday life for counting, measuring, and computing.
• Imaginary Numbers: Not applicable in real-life scenarios; useful for certain mathematical problems (not covered in this course).
• Sets: Collections of distinct objects called elements, typically denoted in braces (e.g., {1, 3, 8, 9, 10}).

Types of Numbers

• Natural Numbers: Counting numbers, all positive integers (e.g., 1, 2, 3...).
• Whole Numbers: Natural numbers plus zero (e.g., 0, 1, 2, ...).
• Rational Numbers: Can be expressed as a fraction of two integers (e.g., 1/2, 0.75).
• Irrational Numbers: Cannot be expressed as a fraction, and their decimal representation is non-terminating and non-repeating (e.g., π, √2).
• Integers: All whole numbers, including negatives, zero, and positives.
• Non-Integers: Numbers that include a fractional part (e.g., 1.5, -3.7).

Functions in Algebra

• A function links inputs (independent variable) to one specific output (dependent variable).
• Functions can be denoted as f(x), where the output depends on the input's value.
• Dependent Variable: Changes based on the independent variable.
• Independent Variable: Input variable that remains constant regardless of changes in other values.

Examples of Functions and Variables

• For a function denoted as f(x), changing x directly alters the output value.
• Numerical calculations require substituting values into the function and simplifying (e.g., solving f(3) by replacing x with 3).

Summary of Number Classification

• Numbers can be categorized into real or imaginary, rational or irrational, integers or non-integers.
• Understanding number types and their classification is fundamental to mastering algebraic principles and operations.

Practical Applications

• Functions model real-world processes (e.g., motion, population growth) and are crucial in data analysis.
• Irrational numbers, like π, are significant in geometric calculations involving circles.

Additional Learning Resources

• Supplementary materials and practice problems are available via Khan Academy for further understanding and reinforcement.### Learning Module Overview
• Module focuses on solving systems of linear equations.
• Key methods to be learned include substitution and elimination methods.

Learning Targets

• Identify various methods for solving linear equations.
• Apply elimination and substitution methods to solve systems of linear equations.

Understanding Expressions, Equations, and Inequalities

• Expressions: Combinations of numbers, variables, and operations without an equals sign (e.g., (3x + 2)).
• Equations: Statements indicating equality between two expressions, represented with an equals sign (e.g., (2x + 3 = 7)).
• Inequalities: Represent relationships of greater than, less than, or equal to (e.g., (x > 5)) and involve flipping the inequality symbol when sides are interchanged.

Linear Equations

• Defined as equations where the variable's highest power is 1, typically represented in the form (ax + b = c).
• Solving involves isolating the variable using inverse operations (addition, subtraction, multiplication, or division).

System of Linear Equations

• A collection of two or more equations with the same variables.
• The goal is to find variable values that satisfy all equations simultaneously.

Methods for Solving Systems

• Substitution Method:

  • Solve one equation for one variable.
  • Substitute this value into another equation to solve for another variable.
  • Verify solutions in both equations to confirm correctness.

• Elimination Method:

  • Eliminate one variable by adding or subtracting the equations.
  • Solve for the remaining variable and substitute back to find others.
  • This method works well when equations are structured to cancel out variables easily.

Example Problem Steps

• Steps to use the substitution and elimination methods were illustrated through detailed examples, demonstrating the processes of solving for variables and verifying solutions.

Summary of Key Concepts

• Expressions and equations are foundational building blocks in algebra.
• Linear equations and systems provide tools for modeling real-world scenarios and solving problems efficiently.
• Both substitution and elimination methods are equally valid; selection depends on the problem's context.
• Always check solutions by substituting back into the original equations to ensure both are satisfied.

Practical Applications

• Understanding systems of linear equations aids in making decisions involving multiple variables in real-life situations, such as budgeting and resource allocation.### Solving Systems of Equations and Word Problems in Algebra
• Systems of equations are expressed in standard form: ( ax + by = c ).
• A system can have no solution, one solution, or infinitely many solutions; this course focuses on systems with one solution.
• Applications of systems of equations include determining ticket sales in a theater through ticket number and revenue equations.

Steps for Solving Word Problems

• Read the Problem Carefully: Understand the context; re-read if necessary.
• Identify the Unknowns: Assign variables (e.g., ( x ) for one quantity, ( y ) for another).
• Extract Key Information: Focus on important numbers and relationships from the problem.
• Translate Words into Equations: Convert verbal descriptions into mathematical equations.
• Set up the Equation: Write down the identified equations based on the established relationships.
• Solve the Equation: Use algebraic methods to find the unknowns.
• Check Your Solution: Substitute back into the original problem to verify accuracy.
• Write the Answer: Clearly state the solution within the context of the problem.

Examples

• Example 1: IT Devices

  • Variables: ( x ) (laptops), ( y ) (tablets).
  • Equations: ( y = 2x ), ( x + y = 18 ).
  • Solution: 6 laptops and 12 tablets.

• Example 2: Two Numbers

  • Variables: ( x ) (smaller number), ( y ) (larger number).
  • Equations: ( y = 3x ), ( x + y = 48 ).
  • Solution: 12 and 36.

• Example 3: School Supplies

  • Variables: ( x ) (cost of pen), ( y ) (cost of notebook).
  • Equations: ( y = x + 5 ), ( 4y + 6x = 60 ).
  • Solution: Pen costs PhP 4, notebook costs PhP 9.

• Example 4: Travel Time

  • Variables: ( t_1 ) (time to Town B), ( t_2 ) (time from Town B).
  • Distance: 120 miles.
  • Equations based on speed and time yield total time of 5 hours.

• Example 5: Teenager's Trip

  • Variables: ( d_1 ) (home to park), ( d_2 ) (park to school).
  • Speeds: 12 km/h (running), 76 km/h (bus).
  • Total distance: 120 km, total time: 2 hours.
  • Solution confirms distances match total traveled.

Summary of Techniques

• Methodical approach is key in solving algebra problems.
• Translating word problems to equations aids in practical applications for real-world scenarios.
• Understanding the relationships between quantities enables effective problem-solving.

Test your understanding of the foundational concepts of algebra in this first module of MAT 204: Mathematics in the Modern World for Engineers. This quiz covers essential algebraic principles that are crucial for engineering applications.