Overview
This set of Algebra 1 lessons covers foundational algebraic concepts, including variables, expressions, equations, properties of equality, solving linear equations (with integers, fractions, and decimals), word problems, ratios, proportions, inequalities, linear equations in two variables, graphing, slope, and special case equations. The goal is to build core algebra skills step by step, preparing students to solve equations and apply algebra to real-world problems.
Variables and Expressions
- A variable is a symbol (usually a letter) representing an unknown value, most often x, y, or z.
- Algebraic expressions combine numbers, variables, and operations (e.g., 2x + 3).
- A coefficient is the number multiplying a variable (e.g., 4 in 4x).
- A constant is a fixed number in an expression (e.g., 7 in 5x + 7).
- Terms are parts of an expression separated by + or − signs.
- Like terms have the same variable with the same exponent (e.g., 3x and 7x).
- Combine like terms by adding/subtracting their coefficients.
Equations and Properties of Equality
- An equation states two expressions are equal (e.g., 2x + 3 = 7).
- The addition property: Add or subtract the same number from both sides to maintain equality.
- The multiplication property: Multiply or divide both sides by the same nonzero number to maintain equality.
- To isolate a variable, undo operations using the opposite operation (e.g., subtract if added, divide if multiplied).
Solving Linear Equations
- Linear equations have the form ax + b = c, where a ≠ 0.
- Solve in these steps:
- Simplify each side (remove parentheses, combine like terms).
- Use addition/subtraction to get the variable term on one side.
- Use multiplication/division to solve for the variable.
- Check your answer by substituting it back into the original equation.
- With fractions, multiply both sides by the least common denominator (LCD) to clear fractions.
- With decimals, multiply both sides by 10, 100, etc., to clear decimals.
Word Problems and Applications
- Assign a variable for the unknown.
- Translate the problem into an equation using the information given.
- Solve for the variable, and state the answer in the context of the problem.
- Common types: sums of quantities, ages, mixture problems, and motion problems (distance = rate × time).
Ratios, Proportions, and Percent
- A ratio compares two quantities (e.g., a:b or a/b).
- A proportion is an equation stating two ratios are equal (a/b = c/d).
- Cross-multiply to solve: a·d = b·c.
- Find unit rates by dividing the numerator by the denominator.
Inequalities
- Inequality symbols: < (less than), ≤ (less than or equal to), > (greater), ≥ (greater or equal).
- Solve linear inequalities like equations, but reverse the inequality if multiplying/dividing by a negative number.
- Graph solutions on a number line or using interval notation.
- Bracket [ ] means endpoint included (≤ or ≥); parenthesis ( ) means not included (< or >).
Linear Equations in Two Variables & Graphing
- Standard form: ax + by = c, where a and b aren’t both zero.
- Ordered pairs (x, y) are solutions to the equation.
- The solution set is graphed as a straight line.
- To graph: find at least two points, plot, and draw a line.
- Slope, m, is rise/run (change in y/change in x): m = (y₂ − y₁)/(x₂ − x₁).
- Slope-intercept form: y = mx + b, where b is the y-intercept (where the line crosses the y-axis).
Special Cases
- A system with no solution: parallel lines (same slope, different y-intercept).
- A system with infinitely many solutions: same line (identical equations).
- Identity: equation true for all values (e.g., 3x − 12 = 3(x − 4)).
- Contradiction: equation never true (e.g., 2 = 5).
Key Terms & Definitions
- Variable — a symbol representing an unknown quantity.
- Coefficient — number multiplying a variable.
- Constant — a fixed value.
- Term — a single variable, a number, or their product.
- Like Terms — same variables raised to the same powers.
- Equation — mathematical statement that two expressions are equal.
- Expression — a number, variable, or combination without an equal sign.
- Inequality — statement showing one quantity is less/more than another.
- Slope — measure of steepness (rise/run) of a line.
- y-intercept — where a line crosses the y-axis.
- Proportion — equation showing two ratios are equal.
- Solution — value(s) that make an equation true.
Action Items / Next Steps
- Practice: Solve equations (including word problems) using both addition and multiplication properties.
- Review: Combine like terms and distribute using the distributive property.
- Homework: Complete problems on solving equations with fractions/decimals.
- Reading: Study examples of graphing linear equations and finding slopes.
- Memorize: Key formulas (distance = rate × time; y = mx + b; a/b = c/d cross-multiplication).
- Prepare for quiz: Practice graphing inequalities, solving word problems, and working with proportions.