Question

If y varies directly with x, and if y = 2 when x = 6, how do you find the direct variation equation?

Answer 1

Hint: In the given question, we have been asked to write the equation of a direct variation and the value of the variable ‘x’ and ‘y’ is given. In order to answer this we need to convert the given proportionality in direct variation equation i.e.\[y=kx\], where k is the value of the constant of the variation. And then putting the values of ‘x’ and ‘y’ we will need to find the value of the constant ‘k’. and we will write the equation by substituting the value of ‘k’.

Complete step-by-step solution:
We have given that,
Y varies directly with x and y = 2 when x = 6.
As it is given that the,
‘y’ is directly proportional to the ‘x’ such that \[y\propto x\].
As we know that,
The equation on converting the proportionality to direct variation equation, i.e.
\[y=kx\], where k is the value of the constant of the variation.
Now,
Putting the values of x = 6 and y = 2, we will obtain
\[\Rightarrow 2=k\times 6\]
Dividing both the sides of the equation by 6, we will obtain
\[\Rightarrow \dfrac{2}{6}=\dfrac{k\times 6}{6}\]
Cancelling out the common factors and simplifying the numbers in the above expression, we get
\[\Rightarrow k=\dfrac{1}{3}\]
Thus,
The required equation of a direct variation will be,
\[\Rightarrow y=\dfrac{1}{3}x\]

Hence, the required equation of a direct variation will be, \[ y=\dfrac{1}{3}x\]

Note: When you have given the question of a direct variation, in direct variation the value of one of the variable changes then the resulting value changes in the same and the proportional manner
A direct variation between x and y can be represented as;
\[y=kx\], where k is the value of the constant of the variation i.e. \[k\in \mathbb{R}\].
The above equation means that if there is an increase in the value of the variable ‘x’ then ‘y’ also tends to increase and if ‘x’ goes smaller then ‘y’ also tends to get smaller.