Question

If y varies directly with x, how do you find y when x=3 if y=-3 when x =6?

Answer 1

Hint: This question is based on the direct variation.
Here are the steps required for solving the direct variation problem:
Write the correct equation. Direct variation problems are solved using the equation $y = kx$.
Use the information given in the problem to find the value of k, called the constant of variation or the constant of proportionality.
Rewrite the equation from step 1 substituting the values of k found in step 2.
Use the equation found in step 3 and the remaining information given in the problem to answer the question asked.

Complete step by step answer:
To solve this question,
The first step is to write the correct equation.
Direct variation problems are solved using the equation below.
$ \Rightarrow y = kx$ ...(1)
The second step is to use the information given in the problem to find the value of k.
In this question, the value of x is 6 and the value of y is -3.
Let us substitute the values in the above equation.
$ \Rightarrow - 3 = k\left( 6 \right)$
Now, let us divide both sides by 6.
We will get,
$ \Rightarrow \dfrac{{ - 3}}{6} = \dfrac{{k\left( 6 \right)}}{6}$
Apply division into both sides.
$ \Rightarrow \dfrac{{ - 1}}{2} = k$
Therefore, the value of k is,
$ \Rightarrow k = - \dfrac{1}{2}$
The third step is substituting the value of k in the equation (1).
$ \Rightarrow y = - \dfrac{1}{2}x$
Now, we want to find the value of y when the value of x is 3.
Therefore, substitute the value of x in the above equation.
$ \Rightarrow y = - \dfrac{1}{2}\left( 3 \right)$
Let us apply multiplication on the right-hand side.
We will get,
$ \Rightarrow y = - \dfrac{3}{2}$

Note: Direct variation: It describes a simple relationship between two variables.
We can write direct variation in maths as:
$ \Rightarrow y \propto x$
We say y varies directly with x.
Let us make an equation by using the constant. It is known as the constant of variation or the constant of proportionality.
Assume the constant is k.
$ \Rightarrow y = kx$
This means that as x increases, y increases, and as x decreases, y decreases. The ratio between them always stays the same.