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Understanding Direct Variation

Sep 14, 2025

Overview

This lecture covers the concept of direct variation, its formula, how to find the constant of variation, and solving for missing variables using examples.

Direct Variation Basics

  • Direct variation describes a relationship where y = kx, with k as the constant of variation.
  • "Y varies directly as x" means y changes in proportion to x; if x increases, y increases, and vice versa.
  • In the equation, y and x are variables and k is always a fixed number.
  • Tables of values or graphs showing both variables increasing indicate direct variation.
  • The graph of a direct variation is a straight line passing through the origin (0,0).

Formula Manipulation

  • The general formula for direct variation is y = kx.
  • To find k, rearrange the formula: k = y / x.

Example Problem 1

  • Given: y = 12 when x = 4.
  • To find k: k = 12 / 4 = 3.
  • Equation of variation: y = 3x.
  • Find x when y = 36: 36 = 3x → x = 36 / 3 = 12.

Example Problem 2 (Table of Values)

  • For x = 3, 5, 7 and y = 6, 10, 14:
    • Calculate k for each pair: k = y / x = 2.
  • General equation: y = 2x.
  • Find y when x = 9: y = 2 × 9 = 18.

Key Terms & Definitions

  • Direct Variation — A relationship where one variable is a constant multiple of another (y = kx).
  • Constant of Variation (k) — The fixed number that relates the two variables in direct variation.
  • Origin — The point (0,0) where the graph of a direct variation passes through.

Action Items / Next Steps

  • Practice finding the constant of variation and writing equations from problem sets.
  • Try solving for missing variables in direct variation problems.
  • Review the concept of graphing direct variation equations.