Question

How do you determine whether the equation represents a direct variation. If it does, find the constant of variation $2y=5x+1$?

Answer 1

From the question we have been asked to find whether the given equation represents the direct variation or not and also if it does, find the constant of variation. We will proceed with the question with the help of definition of the direct variation. Using it we will check whether the given equation is direct variation or not and proceed further calculation.

Complete step by step solution:
Firstly, from the definition of the direct variation we have that, for a given equation if we can bring it in the form of $y=kx$ where k is the constant of variation then the equation is of direct variation and it has a variation constant.
Now, we will simplify the given question and check whether it is in the form of direct variation or not. So, we get,
$\Rightarrow 2y=5x+1$
Now, we will send the two to the other side that is the right hand side of the equation. So, we get the equation reduced as follows.
$\Rightarrow 2y=5x+1$
$\Rightarrow y=\dfrac{5x+1}{2}$
$\Rightarrow y=\dfrac{5x}{2}+\dfrac{1}{2}$
Here, from the above we can clearly say that the given equation in question is in the form of $\Rightarrow y=mx+b$. It is not in the form of $y=kx$.
So, from this we can say that the given equation in the question does not represent a direct variation.
Therefore, we can also say that there is no variation constant for the given equation.

Note: Students must be very careful in doing the calculations. Students should have good knowledge in the concept of direct variation and should know its definition and its applications. Students should not make mistakes like for example if we compare the given equation without reducing it into $y=kx$ form then our solution will be wrong.