1. What Can Algebra Do That Arithmetic Cannot?

Arithmetic solves specific problems. Algebra solves categories of problems — all at once.

• Arithmetic: "What is 5 + 3?" Algebra: "What is x + 3 = 8 for any x?"
• Arithmetic: calculate one answer. Algebra: write a formula for every possible answer.
• Physics formula d = rt works for any distance, rate, and time — one equation, infinite problems
• Financial formulas calculate compound interest for any amount, rate, and time period
• Algebra lets you describe patterns, make predictions, and solve whole families of problems

2. What Is Algebra?

Algebra is the branch of mathematics that uses symbols to represent quantities and relationships, allowing general problems to be solved.

• Variable: a symbol (usually a letter) representing an unknown or changing quantity
• Expression: a combination of numbers, variables, and operations
• Equation: a statement that two expressions are equal
• Function: a rule that assigns one output to each input
• Linear, quadratic, exponential: different types of relationships between variables

3. The History of Algebra

Algebra developed across civilizations over thousands of years before becoming the symbolic language it is today.

• ~1800 BCE — Babylonians solve quadratic equations in words
• 830 CE — Al-Khwarizmi writes the foundational algebra text in Baghdad
• 1500s — European mathematicians develop symbolic notation
• 1637 — Descartes introduces coordinate geometry (graphing equations)
• 1800s–now — Algebra becomes the language of all of science and engineering

4. The Golden Rule of Algebra

Whatever you do to one side of an equation, you must do to the other.

• An equation is a balance: both sides weigh the same
• Adding 5 to the left? Add 5 to the right.
• Multiplying the left by 3? Multiply the right by 3.
• The goal: isolate the variable — get x alone on one side
• Check: substitute your answer back in to verify both sides are equal

5. Properties That Make Algebra Work

Algebra relies on consistent properties of numbers that let you rearrange expressions safely.

• Commutative: a + b = b + a (order does not change sums or products)
• Associative: (a + b) + c = a + (b + c) (grouping does not change sums or products)
• Distributive: a(b + c) = ab + ac (multiply into parentheses)
• Zero product: if ab = 0, then a = 0 or b = 0
• Substitution: if a = b, you can replace a with b anywhere

6. What Algebra I Covers

A roadmap of the major topics in Algebra I.

• Linear equations and inequalities in one variable
• Graphing lines: slope, y-intercept, slope-intercept form
• Systems of linear equations: two equations, two unknowns
• Exponents and polynomials: adding, subtracting, multiplying
• Factoring: undoing multiplication to find roots
• Quadratic equations: parabolas and the quadratic formula

7. Algebra in Real Life

Algebra shows up in every field and in everyday decisions.

• Physics: F = ma (force = mass × acceleration)
• Finance: A = P(1 + r)^t (compound interest)
• Medicine: body surface area formulas for drug dosing
• Engineering: load-bearing calculations for structures
• Sports: calculating batting average, ERA, shooting percentage

8. Solving Algebraically vs. Graphically

Two methods for solving equations — each has strengths.

9. Linear vs. Nonlinear

The most important distinction in Algebra I.

10. Common Algebra Mistakes

The errors that most reliably cause wrong answers.

• Not doing the same operation to both sides: if you add to the left, add to the right
• Sign errors: distributing a negative incorrectly — −2(x − 3) = −2x + 6, not −2x − 3
• Confusing slope and y-intercept in y = mx + b
• Forgetting to check: always substitute your answer back
• Dividing by a variable (you might be dividing by zero)

11. How to Succeed in Algebra

Algebra is learnable by anyone willing to practice consistently.

• Show all steps, always — skipping steps is where errors appear
• Check every answer by substituting back
• Practice daily: 15 minutes is more effective than 2 hours once a week
• When stuck, try a simpler version of the problem first
• Draw a picture or number line when the algebra is confusing

12. Solve for x

Three equations of increasing difficulty — show every step.

• Problem 1: x + 9 = 15 (subtract 9 from both sides)
• Problem 2: 4x − 3 = 17 (add 3, then divide by 4)
• Problem 3: 2(x + 5) = 24 (distribute, then solve)
• For each: write every step, then check by substituting back
• Answers: x = 6, x = 5, x = 7

13. What We Covered

Algebra is the universal language of mathematical relationships — and it is learnable with consistent practice.

• Algebra describes relationships using variables and equations
• The golden rule: whatever you do to one side, do to the other
• Linear equations graph as straight lines; nonlinear equations graph as curves
• Algebra I covers: linear equations, graphing, systems, polynomials, factoring, quadratics
• Succeed by showing all steps, checking answers, and practicing daily

Mastery quiz

1. What is the main goal when you solve an equation?
   • Make both sides as big as possible
   • Isolate the variable so it is alone on one side
   • Erase the equals sign
   • Turn every number into x
2. Which of these is an equation rather than just an expression?
   • 3x plus 5
   • 2 times the quantity x plus 5
   • 3x plus 5 equals 20
   • x
3. Where does the word "algebra" come from?
   • A Greek word meaning "to count"
   • The Arabic "al-jabr," from Al-Khwarizmi's textbook around 830 CE
   • A French word coined by Descartes in 1637
   • An ancient Babylonian word for "balance"
4. What makes a relationship LINEAR, according to the class?
   • It graphs as a curve that grows faster and faster
   • It has a constant rate of change and graphs as a straight line
   • It always involves x squared
   • It can only be solved with a graph, not algebra