Overview
This collection of Algebra 1 lessons covers foundational topics including variables and expressions, equations, properties of equality, linear equations (simple and with fractions/decimals), word problems, proportions, inequalities, linear equations in two variables, graphing, slope, forms of linear equations, systems of equations, polynomials, factoring, rational expressions, roots, the distance formula, and solving quadratic equations by factoring, completing the square, and the quadratic formula.
Variables & Expressions
- A variable is a symbol (often x, y, or z) representing an unknown value.
- An expression combines numbers and/or variables using operations (e.g., 2x + 3).
- A term is a single number, variable, or a product of numbers and variables (e.g., 6, x, 7y).
- The coefficient is the number multiplying a variable in a term.
- A constant is a fixed value that does not change in the expression.
- Algebraic expressions can be evaluated by substituting specific values for variables.
Equations & Properties of Equality
- An equation states that two expressions are equal and always has an “=”.
- To solve, isolate the variable using addition/subtraction and multiplication/division properties.
- Addition Property: Add/subtract the same value on both sides without changing the solution.
- Multiplication Property: Multiply/divide both sides by a nonzero value without changing the solution.
Linear Equations in One Variable
- Standard form is ax + b = c, where a ≠ 0.
- Simplify each side, combine like terms, isolate variable terms on one side and constants on the other.
- To solve: use addition/subtraction then multiplication/division.
- For equations with fractions, multiply both sides by the least common denominator (LCD) to clear fractions.
Word Problems & Applications
- Common word problem types: sums, ages, consecutive integers, mixtures, motion, and rates.
- Steps: Read carefully, define variables, set up equations, solve, then interpret the answer in context.
Proportions & Ratios
- A ratio compares two quantities; can be written as a/b, a:b, or a to b.
- A proportion equates two ratios: a/b = c/d.
- Solve proportions by cross-multiplying: ad = bc.
Inequalities
- Inequality symbols: <, ≤ (less than, less than or equal to); >, ≥ (greater than, greater or equal to).
- To solve: Isolate the variable as with equations.
- When multiplying/dividing by a negative, reverse the inequality sign.
- Solution sets can be expressed using interval notation and graphed on a number line.
Linear Equations in Two Variables & Graphing
- Standard form: ax + by = c.
- Solution is an ordered pair (x, y) making the equation true.
- Graph is a straight line; plot points and draw line through them.
- Slope (m) measures steepness: m = (y₂ - y₁) / (x₂ - x₁).
- Forms: slope-intercept (y = mx + b), point-slope, and standard form.
Systems of Equations
- A system of equations is a set of two or more equations with the same variables.
- Solve by graphing (intersection point), substitution, or elimination.
- Types: one solution (intersect), no solution (parallel), infinite solutions (same line).
Polynomials & Factoring
- Polynomials: expressions with terms of the form axⁿ.
- Add/subtract: combine like terms.
- Multiply: use distributive property or FOIL for binomials.
- Factor by GCF, grouping, trinomials, special products (difference of squares, perfect square trinomials, sum/difference of cubes).
Rational Expressions & Equations
- A rational expression is a ratio of two polynomials.
- To simplify: factor numerator and denominator, then cancel common factors.
- Find restricted values by setting denominators = 0.
- To add/subtract: find the LCD, rewrite with common denominator, then add or subtract numerators.
Roots & Radical Expressions
- The square root, ⁿ√a, is a value which, when raised to n, gives a.
- Principal (positive) root is standard; negative root is written with a minus sign.
- Irrational numbers can’t be written as a fraction of integers; decimal form does not terminate or repeat.
- Simplify radicals by factoring out perfect n-th powers from the radicand.
The Distance Formula
- Distance between two points (x₁, y₁), (x₂, y₂): d = √[(x₂ - x₁)² + (y₂ - y₁)²].
Quadratic Equations
- Standard form: ax² + bx + c = 0, where a ≠ 0.
- To solve by factoring: factor the quadratic and use the zero product property.
- Square root property: if x² = k, then x = ±√k.
- Completing the square: make a perfect square trinomial to use square root property.
- Quadratic formula: x = [-b ± √(b²-4ac)] / (2a); use when factoring is hard or impossible.
Key Terms & Definitions
- Variable — symbol (like x) representing an unknown value.
- Coefficient — the number multiplying a variable in a term.
- Constant — a fixed value.
- Term — a single part of an expression (number, variable, or product).
- Polynomial — an expression of multiple terms combined by + or −.
- Equation — a mathematical statement with an equals sign.
- Inequality — a statement comparing two values with <, >, ≤, or ≥.
- Slope — the measure of the steepness of a line (rise/run).
- Intercept — the point where a graph crosses the x or y axis.
- Factor — a quantity that divides another quantity evenly.
- Quadratic Equation — an equation of the form ax² + bx + c = 0.
- Discriminant — b²−4ac; in the quadratic formula, determines the number of solutions.
- Rational Expression — a ratio of two polynomials.
- Radicand — the value under a radical sign.
- Index (of a radical) — the root being taken (2 for square root, 3 for cube, etc.).
Action Items / Next Steps
- Review and memorize key formulas (quadratic formula, distance, slope, etc.).
- Practice factoring polynomials and solving quadratic equations with a variety of methods.
- Complete assigned textbook exercises including word problems.
- Prepare flashcards for key properties and formulas.
- Familiarize yourself with interval notation and rules for solving inequalities.
- Apply learned problem-solving strategies to new and mixed-type problems.