Question

: How do you find the constant of variation for a direct variation that includes the given values (2, 4)?

Answer 1

Hint: Here we have to find the constant of variation where the points are mentioned or given. usually in a graph the y varies with the value of x and by some constant also. here we are determining that constant value. by the definition of direct variation, we determine the value of constant variation.

Complete step-by-step solution:
The statement "y varies directly as x," means that when x increases, y increases by the same factor. In other words, y and x always have the same ratio and it is given as \[\dfrac{y}{x} = k\]. It can be written as \[y = kx\]where k is a constant of variation.
the equation \[y = kx\] is a direct variation.
let we consider the equation of direct variation as (1) i.e., \[y = kx\]--- (1)
as in the question they have mentioned the points. let us consider the points (2, 4). here the value of x is 2 and the value of y is 4. substituting these values in the equation (1) we get
\[ \Rightarrow 4 = 2k\] ---- (2)
divide the equation (2) by 2
\[ \Rightarrow \dfrac{4}{2} = \dfrac{{2k}}{2}\]
on simplifying we get
\[ \Rightarrow k = 2\]

Therefore the constant variation for a direct variation that includes the given values (2, 4) is 2.

Note: Variation problems involve fairly simple relationships or formulas, involving one variable being equal to one term. The constant of variation in a direct variation is the constant (unchanged) ratio of two variable quantities. In the following equation y varies directly with x, and k is called the constant of variation.