Question

For the direct variation $y = (3/4)$ when $x = (1/8)$ . How do you find the constant of variation and find the value of y when $x = 3$ ?

Answer 1

Hint: This is a variation in graph when a point moves from a location to another in the equation of that graph. So first derive a general equation of this graphical condition and then use that condition to find the value of constant of variation and then substitute that constant’s value and $x = 3$ to find the value of y when $x = 3$ .

Complete step-by-step solution:
This is a question of basic algebra and knowledge of graphs. We know when a graph is linear or straight line then x varies directly as y. That means that when y increases then x increases by the same value or factor. In others we can say that x and y increase or decrease in the same ratio.
So we represent it mathematically, $\dfrac{y}{x} = 1$ . We came to this relation assuming x and y varies linearly, but if we consider a general case then we will have some constant in place of $1$ . So now let us write the same expression as,
$\dfrac{y}{x} = $ some constant, let us say that k is that constant so,
$ \Rightarrow \dfrac{y}{x} = k$ or we can write it as,
\[ \Rightarrow y = kx\] $ - - - - (1)$
Now in the question we are given $y = (3/4)$ when $x = (1/8)$ so if we substitute these values in equation $1$ to get the value of k we get,
$\dfrac{3}{4} = k\left( {\dfrac{1}{8}} \right)$
On evaluating we get,
$ \Rightarrow k = 6$
Now if we substitute $k = 6$ in equation $1$ we will get a general equation of the given condition in the question. So by substituting we get,
\[y = 6x\] $ - - - - (2)$
We got the value of the constant k, now we are asked to find the value of y when $x = 3$ , so by substituting $x = 3$ in equation $2$ , we get
$y = 6 \times 3$
$ \Rightarrow y = 18$
So the constant of variation, $k = 6$ and the value of y when $x = 3$ is $18$.

Note: The general equation formed in this question is an example of a linear equation in two variables which is $ax + by = r$ and here $a = - 6,b = 1,r = 0$ . For solving a linear equation in two variables there are infinitely many solutions because infinite points can be picked up from the graph and substituted in the equation to find its solution.