Question

Is $y = 12x$ a direct variation and if it is, how do you find the constant?

Answer 1

In this question, we need to verify whether the given expression is direct variation or not.
If it is yes, then we need to find out the constant. Firstly, we will verify that y varies directly with x or not. If it's yes, then it’s a direct variation. Then we have the equation for direct variation which is given by, $y = kx$. We compare the given equation with the general form and find out the constant required.

Complete step by step solution:
Given the equation of the form $y = 12x$
We are asked to verify whether the given expression is a direct variation or not.
If it is, then we are asked to find out the constant of variation.
Firstly, let us understand what exactly is direct variation.
Direct variation describes a simple relationship between two variables. We say $y$ varies directly with $x$if, $y \propto x$
i.e. $y = kx$ …… (1)
for some constant k.
The constant k is called the constant of variation or constant of proportionality.
This means that as x increases, the variable y also increases.
On the other hand, as x decreases, then y also decreases.
And that time ratio between them always stays the same.
Note that the graph of the direct variation equation is a straight line through the origin.
We can have k number of straight lines which pass through the origin.
Now let us consider the given equation which is, $y = 12x$
Note that the given equation is in the form of $y = kx$ which is given by the equation (1).
Thus, the given expression is a direct variation.
Now we find the constant of variation.
Now comparing $y = 12x$ with the equation given in (1), we get,
$\Rightarrow k = 12$
Hence the constant of variation is $k = 12$.

Thus, we have proved that $y = 12x$ is a direct variation and the constant of variation is $k = 12$.

Note :
Students should not make any mistakes in such problems, since it is easy to verify. It's just to compare with the general equation and find the required answer.
So remember the formula of direct variation which is given as,
$y = kx$, where k is a constant called variation constant.
For different values of k, we obtain different expressions for y.
Direct variation is nothing but the relation between the two variables in which one is a constant multiple of the other.
For example, when one variable changes the other, then they are said to be in proportion.