💯Math for Non-Math Majors Unit 5 – Algebra

Key Concepts and Definitions

• Algebra branch of mathematics that uses letters and symbols to represent numbers and quantities in equations and formulas
• Variable symbol, usually a letter, used to represent an unknown or changing quantity in an algebraic expression or equation
• Constant fixed value that does not change in an algebraic expression or equation
• Coefficient number multiplied by a variable in an algebraic term (e.g., in the term $3x$, the coefficient is 3)
• Exponent number that indicates how many times a quantity is multiplied by itself (e.g., in the term $x^2$, the exponent is 2)
• Term single mathematical expression that may include numbers, variables, and exponents connected by multiplication or division
• Like terms terms in an algebraic expression that have the same variables and exponents, and can be combined by adding or subtracting their coefficients
• Equation mathematical statement that shows two expressions are equal, using the equals sign ($=$)
  • Solving an equation finding the value of the variable that makes the equation true

Basic Algebraic Operations

• Addition combining like terms by adding their coefficients while keeping the variable and exponent the same
• Subtraction combining like terms by subtracting their coefficients while keeping the variable and exponent the same
• Multiplication multiplying coefficients and adding exponents when multiplying terms with the same base
  • Distributive property multiplying a factor outside parentheses by each term inside the parentheses ($a(b+c) = ab + ac$)
• Division dividing coefficients and subtracting exponents when dividing terms with the same base
• Exponents raising a base to a power, which is equivalent to multiplying the base by itself the number of times indicated by the exponent
  • Negative exponents indicate the reciprocal of the base raised to the positive exponent ($x^{-n} = \frac{1}{x^n}$)
• Order of operations (PEMDAS) performing algebraic operations in the correct order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Simplifying expressions combining like terms and applying the order of operations to reduce an algebraic expression to its simplest form

Solving Equations and Inequalities

• Solving equations finding the value of the variable that makes the equation true
  • Isolate the variable on one side of the equation by performing the same operation on both sides
• Solving inequalities finding the range of values that make the inequality true
  • Isolate the variable on one side of the inequality by performing the same operation on both sides
  • Flip the inequality sign when multiplying or dividing by a negative number
• Linear equations equations that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants and $a \neq 0$
• Quadratic equations equations that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$
  • Quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ used to solve quadratic equations
• Systems of equations two or more equations with the same variables that are solved simultaneously
  • Substitution method solving one equation for a variable and substituting the result into the other equation
  • Elimination method multiplying equations by constants to eliminate one variable when the equations are added or subtracted

Graphing and Coordinate Systems

• Coordinate plane two-dimensional surface formed by the intersection of a horizontal number line (x-axis) and a vertical number line (y-axis) at their zero points (origin)
• Ordered pair (x, y) represents a point on the coordinate plane, where x is the horizontal coordinate and y is the vertical coordinate
• Quadrants four regions of the coordinate plane divided by the x-axis and y-axis (I, II, III, and IV)
• Graphing representing equations and inequalities visually on the coordinate plane
  • x-intercept point where a graph crosses the x-axis (y = 0)
  • y-intercept point where a graph crosses the y-axis (x = 0)
• Slope measure of the steepness of a line, calculated as the change in y divided by the change in x ($m = \frac{\Delta y}{\Delta x}$)
  • Positive slope line rises from left to right
  • Negative slope line falls from left to right
  • Zero slope horizontal line
  • Undefined slope vertical line
• Equation of a line $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept

Real-World Applications

• Proportions equations stating that two ratios are equivalent ($\frac{a}{b} = \frac{c}{d}$)
  • Can be used to solve problems involving similar triangles, scale drawings, and unit conversions
• Percent equations equations involving percentages, often used in financial calculations (sales tax, discounts, interest rates)
  • Percent change $\frac{\text{new value} - \text{original value}}{\text{original value}} \times 100\%$
• Mixture problems problems involving combining two or more substances with different concentrations to create a mixture with a desired concentration
  • Can be solved using a system of equations
• Distance-rate-time problems problems involving the relationship between distance traveled, rate (speed), and time
  • Distance = Rate × Time
• Optimization problems problems that involve finding the maximum or minimum value of a function subject to certain constraints
  • Can be solved using algebra and calculus techniques

Common Pitfalls and How to Avoid Them

• Sign errors incorrectly applying the rules for adding, subtracting, multiplying, or dividing positive and negative numbers
  • Double-check signs when performing operations and simplifying expressions
• Forgetting to distribute multiplying only the first term inside parentheses when using the distributive property
  • Multiply each term inside the parentheses by the factor outside the parentheses
• Incorrect order of operations performing operations in the wrong order, leading to incorrect results
  • Remember PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Misusing the equal sign writing statements that are not true equations
  • Ensure that the expressions on both sides of the equal sign are equivalent
• Dividing by zero attempting to divide a number by zero, which is undefined
  • Check for potential division by zero when solving equations and simplifying expressions
• Misinterpreting graphs incorrectly identifying key features (x-intercepts, y-intercepts, slope) or confusing the x and y variables
  • Carefully label axes and double-check the interpretation of the graph
• Rounding errors introducing inaccuracies when rounding decimal answers prematurely
  • Carry out calculations with full precision and round only the final answer when necessary

Study Tips and Tricks

• Practice, practice, practice the more problems you solve, the more comfortable you will become with algebraic concepts and techniques
  • Work through a variety of problems, from basic to challenging, to build your skills and confidence
• Show your work writing out each step helps you organize your thoughts, avoid mistakes, and identify areas where you need more practice
  • Use this to your advantage when reviewing your work or seeking help from others
• Check your answers substitute your solution back into the original equation or inequality to verify that it makes the statement true
  • If the answer doesn't check, retrace your steps to find and correct any errors
• Learn from your mistakes review incorrect problems and focus on understanding why you made the error and how to avoid it in the future
  • Identify patterns in your mistakes and develop strategies to prevent them
• Seek help when needed don't hesitate to ask your teacher, tutor, or classmates for clarification or guidance when you're stuck
  • Collaborating with others can provide new perspectives and deepen your understanding
• Create a study schedule break your study sessions into manageable chunks and set specific goals for each session
  • Consistent, focused studying is more effective than long, infrequent cram sessions
• Use memory aids create flashcards, mnemonics, or summary sheets to help you remember key concepts, formulas, and techniques
  • Regularly review these aids to reinforce your learning and maintain your skills

Additional Resources

• Khan Academy free online platform offering a wide range of algebra lessons, practice problems, and instructional videos
  • Covers topics from basic algebraic concepts to advanced techniques
• Wolfram Alpha computational knowledge engine that can provide step-by-step solutions to algebraic problems
  • Useful for checking your work and exploring alternative solution methods
• Desmos online graphing calculator that allows you to graph equations, plot points, and explore the behavior of functions
  • Helps develop a visual understanding of algebraic concepts
• Purplemath free online algebra tutorials and practice problems, organized by topic
  • Includes detailed explanations and worked examples
• MathPapa free online algebra calculator that provides step-by-step solutions to linear and quadratic equations, inequalities, and systems of equations
  • Helps you check your work and understand the problem-solving process
• Your textbook and class notes valuable resources that provide explanations, examples, and practice problems tailored to your course
  • Review these materials regularly to reinforce your learning and identify areas for improvement
• Study groups collaborating with classmates can help you learn from others, share ideas, and stay motivated
  • Organize regular study sessions to review concepts, discuss problems, and prepare for exams