Question

How do you determine whether y varies directly with x and if so, how do you find the constant of variation k for \[ - 5y = 5x + 10\] ?

Answer 1

Hint: Here in this question solve by using a Direct variation method, taking 5 as common in the given equation and divide both side by 5, then compare the equation to the expression y=kx using this we can determine how y varies directly with x and also find the constant of variation k

Complete step-by-step answer:
The Direct Variation states that "y varies directly as x," means that when x increases, y increases by the same factor. In other words, y and x always have the same ratio:
  \[\dfrac{y}{x} = k\]
Where k is the constant of variation.
We can also express the relationship between x and y as :
  \[y = kx\]
where k is the constant of variation.
Since k is constant (the same for every point), we can find k when given any point by dividing the y-coordinate by the x-coordinate.
Consider, the given equation
  \[ \Rightarrow - 5y = 5x + 10\]
Take 5 common on RHS side, then
  \[ \Rightarrow - 5y = 5\left( {x + 2} \right)\]
Divide both side by 5, then
  \[ \Rightarrow - \dfrac{5}{5}y = \dfrac{5}{5}\left( {x + 2} \right)\]
  \[ \Rightarrow - y = x + 2\]
Multiply -1 on both sides,
  \[ \Rightarrow y = - x - 2\]
By definition of Direct variation mathematical relationship between two variables that can be expressed by an equation in which one variable is equal to a constant times the other
from which it follows that the given equation is not a direct variation (i.e., y does not vary directly with x)
The equation of a line parallel to this which would be a direct variation would be
  \[ \Rightarrow y = - x\]
Or
  \[ \Rightarrow y = \left( { - 1} \right)x\]
and for this equation the constant of variation k would be (-1)
Hence, in the equation \[ - 5y = 5x + 10\] , y does not vary directly with x and the constant of variation k is (-1).

Note: The graph is plotted x-axis versus y axis. The graph is two dimensional. By the equation of a graph, we can plot the graph by assuming the value of x. We can’t assume the value of y. because the value of y depends on the value of x. sometimes the y depends on the value of x and constant which is present in the equation.