Question

Is $ 3x + y = 6 $ a direct variation and if it is, how do you find the constant?

Answer 1

Hint: We have to first check whether the given equation is a direct variation. A direct variation between two variables is established when the value of one variable increases with increase in the value of the other variable and decreases with the decrease in the value of the other variable. In other words, a direct variation is established when the two variables are directly proportional.
 $
  y \propto x \\
   \Rightarrow y = kx \;
 $
where $ k $ is the variability constant and $ k > 0 $ .

Complete step by step solution:
We have been given an equation $ 3x + y = 6 $ . We have to check whether this equation is a direct variation or not.
Direct variation means that the value of one variable will increase as we increase the value of the other variable and vice-versa.
To find this we try to write the equation with $ y $ on one side and all other terms on the other side, i.e. we are writing $ y $ in terms of $ x $ .
 $
  3x + y = 6 \\
   \Rightarrow y = - 3x + 6 \;
  $
We get the equation in the form of $ y = mx + c $ where $ m = - 3 $ and $ c = 6 $ .
The value of the coefficient of $ x $ is a negative number and intercept is non-zero.
Also, we can observe that the value of $ y $ will decrease as we increase the value of $ x $ in the equation.
Thus, the given equation is not a direct variation.

Note: To check whether or not the given equation is a direct variation we observed the coefficient of $ x $ . A negative coefficient means inverse variation. A direct variation is established when $ \dfrac{y}{x} = k $ where $ k > 0 $ . Also, when the intercept $ c \ne 0 $ it is partial variation and not direct variation.