Question

If $y$ varies directly as $x$. And $y=12$ when $x=4$, then find the linear equation.

Answer 1

Hint: As y is varying directly with $x$, we can say that y is directly proportional to $x$. Now, we can obtain the linear equation by removing the proportionality sign and adding an arbitrary constant $k$ on LHS. Now, to find the value of $k$ substitute $x=4$ and $y=12$ in the obtained equation.

Complete step by step solution:
In this question, we are given that $y$ is varying directly with x when the value of $x$ is $4$, the value of $y$ will be $12$. Based on this, we need to find the linear equation.
As $y$ directly varies with $x$, we can say that $y$ is directly proportional to $x$. That means,
$y\propto x$
Now, we can remove the proportionality sign by equating $y$ with $x$ and adding an arbitrary constant $k$ on the LHS. Therefore,
$\Rightarrow y=kx$
This is our required linear equation and now we need to find the value of $k$. For that we are given that the value of $y$ is $12$ when the value of $x$ is $4$. Therefore, substituting these values in the above equation, we get
$$\begin{gathered} \Rightarrow 12=4k \\ \Rightarrow k={\displaystyle \frac{12}{4}} \\ \Rightarrow k=3 \end{gathered}$$
Therefore, the required linear equation is $y=3x$.

Note:
Here, note that the relation between y and x was varying directly and so we took $y\propto x$. But if the relation was given as y is varying inversely with $x$, then we had to take $y\propto {\displaystyle \frac{1}{x}}$ and on removing the proportionality sign, we get
$\Rightarrow y=k\cdot {\displaystyle \frac{1}{x}}$