Question

If $y$ varies directly with $x$ and $y=21$ when $x=7$, how do you find $y$ when $x=-1.66$?

Answer 1

Hint: In this question we have been given that $y$ and $x$ which are two variables, vary with each other. This means the change in the value of $1$ value would result in the change of another value. Since it is mentioned in the question that $y$ varies directly with $x$, we will use the rule of directly proportional variables and write it in the form of an equation. We will then find the value of $k$ which is a constant and solve for the value of $x=-1.66$ to get the required solution.

Complete step by step answer:
We know that $y$ varies directly with $x$, this means that $x$ and $y$ are directly proportional to each other. Mathematically it can be written as:
$\Rightarrow y\propto x$
Now the rule of proportion states that if we want to remove the directly proportional sign and write the $=$ sign, we have to multiply the terms with a constant. Consider the constant to be $k$, the equation can be written as:
$\Rightarrow y=kx$
Now we have $y=21$ when $x=7$. On substituting the values in the expression, we get:
$\Rightarrow 21=k\times 7$
On rearranging the terms in the expression, we get:
$\Rightarrow k=\dfrac{21}{7}$
On simplifying, we get:
$\Rightarrow k=7$
So, our formula becomes $y=3x$.
Now when $x=-1.66$, we get the value as:
$\Rightarrow y=3\times -1.66$
On multiplying, we get:
$\Rightarrow y=-4.98$, which is the required solution.

Note: In this question we were given with two variables which were directly proportional to each other. There can also exist a case when two variables are inversely proportional to each other. Mathematically it can be expressed as $y\propto \dfrac{1}{x}$. And when the proportion sign is removed, the equation is in the format $y=\dfrac{k}{x}$, where $k$ is the constant of proportionality.