Question

How do you determine the constant of variation for the direct variation given by $ \left( {1,8} \right)\left( {2,4} \right)\left( {4,2} \right)\left( {8,1} \right) $ ?

Answer 1

Hint: In this problem we asked to determine the constant of variation for the direct variation given by the direct variation set. And also we have to find the given set represents a direct variation or inverse variation with a constant. The constant of variation can be finding by using some formulas

Formula used:
The formula for the direct variation is $ y = c.x $ . The formula for inverse variation is $ x.y = c $ , where $ y $ varies directly as $ x $ , $ y $ is directly proportional to $ x $ and $ c $ is the constant.

Complete step by step answer:
Given set is $ \left( {1,8} \right)\left( {2,4} \right)\left( {4,2} \right)\left( {8,1} \right) $
We have to find the constant of variation.
The constant of variation in a direct variation is the constant ratio of two variable quantities.
If $ \left( {x,y} \right) \in $ direct variation set, then we can write $ y = c \times x $ for some constant $ c $ and all pairs $ \left( {x,y} \right) $ .
If we consider only the first two pairs of the defining set, we have,
 $ 8 = c \times 1 - - - - - (1) $
And $ 4 = c \times 2 - - - - - (2) $
Obviously both cannot be true for any single value of $ c $ . If we give the same value for $ c $ in equation (1) and in equation (2) then both the equations are never satisfied.
However, if $ y $ varies inversely with $ x $ then $ x \times y = c $ for some constant $ c $ and all pairs $ \left( {x,y} \right) $ .
When we considering the first pair, $ \left( {1,8} \right) \to 1 \times 8 = 8 $
When we considering the second pair, $ \left( {2,4} \right) \to 2 \times 4 = 8 $
When we considering the third pair $ \left( {4,2} \right) \to 4 \times 2 = 8 $
When we considering the fourth pair $ \left( {8,1} \right) \to 8 \times 1 = 8 $
So it is clear that it is an inverse variation with a constant of $ 8 $ .

Therefore, the given set does not represent a direct variation; it is an inverse variation with a constant of $ 8 $ .

Note: For a direct variation if we multiply the value of one of our variables by a number, it must result in the value of the other variable being multiplied by that same number. For an inverse variation if we multiply the value of one of our variables by a number, it must result in the value of the other variable being divided by that number.