Question

How do you write the direct variation equation if y varies directly with x and y=10 when x=2?

Answer 1

This type of problem is based on the concept of proportionality. First, we have to consider y varies directly with x, which means $y\text{ }\propto \text{ }x$. Here, $\propto$ is proportional. Then, substitute a proportionality constant k, that is y=kx. And we have been given that y=10 when x=2. On substituting this value, we get the proportionality constant k and thus find the variation equation.

Complete step by step answer:
According to the question, we are asked to find the direct variation equation if y varies directly with x and y=10 when x=2.
We have been given that y varies directly with x.

We first have to consider that y varies directly with x.
This means that y is directly proportional to x.
$y\text{ }\propto \text{ }x$ ------(1)
To remove the proportionality, we can substitute a proportionality constant to the equation (1).
We get,
$y=kx$ ------(2)
Here, k is the proportionality constant.
And we have been given that y=10 when x=2.
Let us now substitute these values in the equation (2).
We get,
10=k(2)
But we know that 10=2x5.
Therefore, we get
2x5=k(2)
Let us now divide 2 on both the sides of the obtained equation.
We get,
$\dfrac{2\times 5}{2}=k\times \dfrac{2}{2}$
On further simplification, we get,
$5=k\times 1$
$\Rightarrow 5=k$
$\therefore k=5$
We have now found the value of the proportionality constant k.
On substituting the value of k in equation (2), we get
$y=5x$
Therefore, the direct variation equation is $y=5x$.

Hence, the direct variation equation if y varies directly with x and y=10 when x=2 is $y=5x$.

Note: Whenever you get this type of problem, we should always try to make the necessary calculations in the given equation to get the final answer. We should avoid calculation mistakes based on sign conventions. We should not forget to put proportionality constant. We may even be asked to find the value of x when y=5 using the direct variation equation.