Algebra 1 Unit 5 Study Guide

Algebraic Forms and Definitions

• Standard Form: Equation of a line represented as Ax + By = C, where A, B, and C are integers.
• Slope-Intercept Form: Expresses the equation of a line as y = mx + b, where m is the slope and b is the y-intercept.
• Point-Slope Form: Formulated as y - y1 = m(x - x1), allows calculation of the line's equation using a point (x1, y1) and slope m.

Slope Concepts

• Slope Formula: Calculated as m = (y2 - y1) / (x2 - x1), indicating the steepness of a line.
• Slope of Zero: Represents a horizontal line, mathematically expressed as 0/1.
• Undefined Slope: Represents a vertical line, mathematically expressed as 1/0.

Graphical Representations

• Horizontal Line Example: The equation y = 1 indicates every y value is 1, resulting in a horizontal line.
• Vertical Line Example: The equation x = 2 means every x value is -2, leading to a vertical line.

Writing Linear Equations

• To create an equation using slope and y-intercept:
  • Slope: 2/7, y-intercept: -3 results in y = (2/7)x - 3.
  • Slope: 0.8, y-intercept: 10 results in y = 0.8x + 10.

Real-World Situations Modeled as Linear Equations

• Television Repair Shop: Charges $35 plus $20 per hour modeled by y = 20x + 35.
• Melting Icicle: Starts at 12 inches, melts at 1/4 inch/hour, modeled by y = 12 - (1/4)x.
• Temperature Increase: Begins at 38 degrees, rises at 3 degrees/hour, resulting in y = 3x + 38.
• Taxi Fare: Charges $5 plus $0.30 per mile, modeled by y = 0.3x + 5.

Finding Inverses

• The inverse of the set {(1, 2), (3, -4), (-7, 9), (8, 8)} is found as {(2, 1), (-4, 3), (9, -7), (8, 8)}.