Direct Variation

Definition

• Direct variation explains the relationship between two variables, where:

  • If variable ( y ) increases, then variable ( x ) also increases at the same rate, and vice versa.
  • This relationship is expressed as ( y ) varies directly as ( x ).

• General equation for direct variation:

  • ( y ) varies directly as ( x^n ) can be written as:
    • ( x \propto x^n ) (variation relation)
    • ( x = kx^n ) (equation relation)
  • Where ( n = 1, 2, 3, \frac{1}{2}, \frac{1}{3} ) and ( k ) is a constant.

Example

• Given ( m = 12 ) when ( n = 3 ).
• Express ( m ) in terms of ( n ):

Case a: ( m ) varies directly as ( n )

• ( m = kn )
• Substitute ( m = 12 ) and ( n = 3 ) into the equation:
  • ( 12 = k(3) )
  • ( k = \frac{12}{3} = 4 )
• Therefore, ( m = 4n ).

Case b: ( m ) varies directly as ( n^3 )

• ( m = ln^3 )
• Substitute ( m = 12 ) and ( n = 3 ) into the equation:
  • ( 12 = l(3)^3 )
  • ( l = \frac{12}{27} = \frac{4}{9} )
• Therefore, ( m = \frac{4}{9}n^3 ).

Related Topics

• Inverse Variation
• Joint Variation
• Matrices
• Basic Operation on Matrices
• Risk and Insurance Coverage
• Taxation
• Congruency
• Enlargement
• Combined Transformation
• Tesselation

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