Answer
a. $y$ does vary directly with $x$.
The constant of variation is $-\frac{5}{3}$.
b. $y$ does vary directly with $x$.
The constant of variation is $\frac{1}{9}$.
Work Step by Step
If $y$ varies directly with $x$, then the equation would take the form $y = kx$, where $k$ is the constant of variation.
To see if these equations are direct variation equations, rewrite the equations such that $y$ is on the left-hand side.
a. Subtract $5x$ from each side of the equation:
$$3y = -5x$$
Divide each side of the equation by $3$ to isolate $y$:
$$y = -\frac{5}{3}x$$
We can see that the equation does take the form $y= kx$; therefore, $y$ does vary directly with $x$.
The constant of variation, $k$, is the coefficient of the variable $x$. In this case, $k$ is $-\frac{5}{3}$.
b. This equation can be rewritten so that all terms can be seen more clearly:
$$y = \frac{1}{9}x$$
We can see that the equation does take the form $y= kx$; therefore, $y$ does vary directly with $x$.
The constant of variation, $k$, is the coefficient of the variable $x$. In this case, $k$ is $\frac{1}{9}$.
