   SEARCH HOME Math Central Quandaries & Queries  Question from liz, a student: how do you tell if an equation varies Directly or Inversely? ex. x= 2y or xy+12=3 i know that if you have three numbers (x. y, z) that it varies jointly but how do you tell if it is Inverse or Direct? Thanks for the Help! -liz Hi Liz.

A direct variation is one where if one quantity goes up, the other quantity goes up and vice versa.

An inverse variation is one where one goes down as the other goes up.

Generally when you see the term xy, you are dealing with an inverse relationship. For example: xy = 2. If x is 10, then y must be 1/5. If x is 100, then y must be 1/50. If x is 1, then y is 2 and if x is 1/1000, then y is 2000.

When things vary directly, then they change size the same way. If x=2y, then when x=5, y=10. If we double x, we double y (10 and 20) and if we reduce to x=1, then y = 1/2.

Notice that it is the magnitude of the size, not strictly greater/less than here. For example, the equation
x = -2y is also a direct variation, but as x gets further away from zero, y gets twice as far from zero in the opposite direction.

Stephen La Rocque.>     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.