Question

If y varies directly as x, and y = 42 as x = 6, how do you find y for the x-value 3?

Answer 1

Hint: The given problem is based on direct variation. We know that y varies directly as x and we have to find the value of x for the given value of y. To understand the concept of direct variation, it is described as a simple variation between two variables, we can say that y varies directly with x, that is\[\dfrac{x}{y}=k\] for some constant k.

Complete step by step answer:
We know that the direct variation can be represented as,
\[\dfrac{x}{y}=k\] ……. (1)
Where k is the constant and y varies directly as x.
We also know that the given value of x and y is,
x = 6 and y = 42, substituting the value of x and y in (1), we get
\[\begin{align}
  & \Rightarrow k=\dfrac{6}{42} \\
 & \Rightarrow k=\dfrac{1}{7} \\
\end{align}\]
Here, we have the value of k.
To find the value of y, for the x-value 3 and k value \[\dfrac{1}{7}\].
Substituting the x and k values in (1), we get
\[\Rightarrow \dfrac{3}{y}=\dfrac{1}{7}\]
Here, we can cross multiply to get the value of y,
\[\Rightarrow y=21\]
Therefore, the value of y for x = 3 is 21.

Note: Students make mistakes in understanding the concept of direct variation which is to be concentrated. We can also find the value of y by observing the other values which varies directly.
By observing the values, we can see that y varies directly as x.
When x = 6, y = \[7\times 6\] = 42
When x = 3, y = \[7\times 3\] = 21
From this we can know that, y is 7 times x, as the constant k = 7.