Variations in Mathematics: Direct, Inverse, and Joint Variation

Introduction to Variations

• Types of variations:
  • Direct Variation
  • Inverse Variation
  • Joint Variation

Direct Variation

• Definition: If y varies directly with x, the equation is:
  • y = kx
• Examples:
  • If R varies directly with S: R = kS
  • If Z varies directly with L: Z = kL

Inverse Variation

• Definition: If y varies inversely with x, the equation is:
  • y = k/x
• Examples:
  • If R varies inversely with S: R = k/S
  • If Z varies inversely with L: Z = k/L

Joint Variation

• Definition: If y varies jointly with x and z, the equation is:
  • y = kxz
• Examples:
  • If y varies jointly with R and S: y = kRS

Combining Variations

• Direct with Inverse:
  • Example: If y varies directly with x but inversely with z:
    • y = kx/z
• Direct with square:
  • Example: If y varies directly with the square of x:
    • y = kx²
• Inverse with square root:
  • Example: If y varies inversely with the square root of R:
    • y = k/√R
• Direct with cube, inverse with square:
  • Example: y = kx³/z²

Solving Word Problems

Example 1: Direct Variation

• Problem: If y varies directly with x, and if x = 3 when y = 12, find y when x = 9.
  • Equation: y = kx
  • Solve for k:
    • 12 = k(3) → k = 4
  • Find y when x = 9:
    • y = 4(9) = 36

Example 2: Inverse Variation

• Problem: If x = 4 when y = 48, find y when x = 8.
  • Equation: y = k/x
  • Solve for k:
    • 48 = k/4 → k = 192
  • Find y when x = 8:
    • y = 192/8 = 24

Example 3: Joint Variation

• Problem: If y = 36 when x = 2 and z = 3, find y when x = 4 and z = 6.
  • Equation: y = kxz
  • Solve for k:
    • 36 = k(2)(3) → k = 6
  • Find y when x = 4 and z = 6:
    • y = 6(4)(6) = 144

Example 4: Direct with Square of x

• Problem: If y = 8 when x = 2, find y when x = 4.
  • Equation: y = kx²
  • Solve for k:
    • 8 = k(2²) → k = 2
  • Find y when x = 4:
    • y = 2(4²) = 32

Example 5: Inverse with Cube of x

• Problem: If y = 108 when x = 2, find y when x = 6.
  • Equation: y = k/x³
  • Solve for k:
    • k = 108(2³) = 864
  • Find y when x = 6:
    • y = 864/(6³) = 4

Example 6: Direct with Cube, Inverse with Square

• Problem: If y = 8 when x = 2 and z = 3, find y when x = 4 and z = 6.
  • Equation: y = k(x³/z²)
  • Solve for k:
    • k = (8*9)/8 = 9
  • Find y when x = 4 and z = 6:
    • y = 9(4³/6²) = 16*

Conclusion

• Understanding how to identify and write equations for direct, inverse, and joint variation is crucial in solving related word problems. Use both conceptual and algebraic methods to find solutions.