Question

How do you write a direct variation equation that relates x and y if y = -8 when x = 2, and how do you find y when x = 32?

Answer 1

Hint: Here, the given problem is based on the direct variation. When we have a direct variation, we can say that as the variable changes, the resulting value changes in the same and proportional manner. A direct variation between y and x is typically denoted by \[y=k x\], where k belongs to Real numbers. This means when x goes smaller y also tends to be smaller and vice versa. Here in this problem, by substituting the value of \[\left( x ,y \right)\] in the direct variation we get the value of k. the value of y can be found by substituting the value of k and given x = 32.

Complete step by step answer:
We know that, the direct variation between y and x is denoted by
\[y=k x\]……. (1)
Where the constant k belongs to real numbers.
We also know that the given point \[\left( x, y \right)\]is \[\left( 2,-8 \right)\]
Substituting the value of x and y in the direct equation (1), We get,
\[\begin{align}
  & \Rightarrow \left( -8 \right)=k\left( 2 \right) \\
 & \Rightarrow k=\dfrac{-8}{2} \\
 & \Rightarrow k=-4 \\
\end{align}\]
Here, we found the constant value, k = -4.
Now we have to find y when x = 32,
Here, in the direct variation equation (1), we can substitute the value of k and given x = 32 to find y
We get,
\[\begin{align}
  & \Rightarrow y=\left( -4 \right)32 \\
 & \Rightarrow y=-128 \\
\end{align}\]

Therefore, the value of y is -128.

Note: Students may get confused about two values of x, here in this problem the first value of x is the point, which satisfies the required equation. The second value of x is to find the value of y from the direct variation equation. It is required to understand the concept of direct variation equation for these types of problems.