Question

Each pair of values is from direct variation, how do you find the missing value for \[\left( {3,7} \right)\] , \[\left( {8,y} \right)\] \[?\]

Answer 1

Hint: To find the missing value for \[\left( {3,7} \right)\] , \[\left( {8,y} \right)\] when each pair of values is from direct variation. We have to find the value of the constant of proportionality using a point. After that substitute the value of the constant of proportionality in direct variation. Then find the missing value using direct variation after substituting the constant of proportionality.

Complete step-by-step answer:
Direct variation means as your independent variable changes, the resulting dependent variable changes in the same and proportional manner. It means that as the independent variable increases(or decreases), the resulting dependent variable is increasing (or decreasing).
A direct variation between \[y\] and \[x\] is denoted by \[y = tx\] where \[t \in \mathbb{R}\] . -----(1)
It means that as \[x\] is increasing (or decreasing), \[y\] also increasing (or decreasing).
First find \[t\] using the given point \[\left( {3,7} \right)\]
Substitute the values \[x = 3\] and \[y = 7\] in the equation (1), we get
 \[7 = 3t\]
 \[ \Rightarrow t = \dfrac{7}{3}\]
Hence, substituting the value of \[t\] in the equation (1), we get
 \[y = \dfrac{7}{3}x\] ----(2)
Since we have to find the value \[y\] of at \[x = 8\] .
Put \[x = 8\] in the equation (2), we get
 \[y = \dfrac{{56}}{3}\] .
So, the correct answer is “ \[y = \dfrac{{56}}{3}\] .”.

Note: In inverse variation, as an independent variable increases (or decreases), the resulting dependent variable is decreasing (or increasing). Inverse variation between \[y\] and \[x\] is denoted by \[y = \dfrac{t}{x}\] where \[t \in \mathbb{R}\] . Hence, the constant of proportionality is the ratio between two quantities that are directly proportional.