Question

If y varies directly as x and y = -24 when x = 8, then how do you find y when x = 10?

Answer 1

Hint: Here in this question, based on the Inverse Variation Models. Given y varies directly as x means y is directly proportional to the k. We have to find the value of y when the value of x = 10 by adding proportionality constant in the place of proportional sign and further simplify using the basic arithmetic operation to get the required solution.

Complete step by step answer:
Inverse variation formula refers to the relationship of two variables in which a variable increases in its value, the other variable decreases and vice-versa. In other words, the inverse variation is the mathematical expression of the relationship between two variables whose product is a constant.
Here in the given question,
Consider the initial statement i.e., if y varies directly as x
In mathematically we represent the initial stamen as
\[ \Rightarrow \,\,\,y\,\alpha \,x\]----------(1)
To convert above equation by multiplying k or replace proportional sign by k, then we have
\[ \Rightarrow \,\,\,y\, = k\,x\]----------(2)
Where, k is proportionality constant
Now, to find the value k use the above condition i.e.,
\[ \Rightarrow \,\,\,y\, = k\,x\]
Divide both side by x, then
\[ \Rightarrow \,\,\,\dfrac{y}{x} = k\,\]
By substituting the value of y = -24 when x = 8, we get the value ok k or value of proportionality constant
\[ \Rightarrow \,\,\,k = \dfrac{{ - 24}}{8}\]
On simplification, we get
\[ \Rightarrow \,\,\,k = - 3\]
Substitute k value in equation (2), we have
\[ \Rightarrow \,\,\,y\, = - 3x\]-----------(3)
Now, we have to find the value of y when x = 10 on substituting in equation (3), we get
\[ \Rightarrow \,\,\,y\, = - 3\left( {10} \right)\]
\[ \Rightarrow \,\,\,y\, = - 30\]
Hence this then means that, when value x increases, value y will decrease, since its inversely proportional.

Note: For solving this type question we have to know the difference between direct and inverse variation. When two things x and y vary directly, it means that as x goes larger, y also goes larger. And when x goes smaller, y also goes smaller
On the other hand, when two things u and v vary inversely, it means that as u goes larger, v goes smaller. And when u goes smaller, v goes larger.