Question

How can I find the domain and range of $2x-3y=6$?

Answer 1

Hint: In the given equation $2x-3y=6$, there are two variables x and y. We can choose x as the independent variable and y as the dependent variable and separate y in terms of x. For this, we have to subtract $2x$ from both sides of the given equation to get the equation $-3y=6-2x$. Then dividing the obtained equation by $-3$ we will get the equation $y=\dfrac{2}{3}x-2$. The set of the allowed values of x which can be put into the equation $y=\dfrac{2}{3}x-2$ will be the domain, and the set of the corresponding values of y obtained will be the range.

Complete step by step solution:
The equation given in the above question is
$\Rightarrow 2x-3y=6$
Let us choose x as the independent variable, and y as the dependent variable. So we try to obtain y in terms of x. For this, we subtract $2x$ from both sides of the above equation to get
$\begin{align}
  & \Rightarrow 2x-3y-2x=6-2x \\
 & \Rightarrow -3y=6-2x \\
\end{align}$
Dividing both sides by $-3$, we get
$\begin{align}
  & \Rightarrow \dfrac{-3y}{-3}=\dfrac{6-2x}{-3} \\
 & \Rightarrow y=\dfrac{2}{3}x-2 \\
\end{align}$
Now, we know that the domain is a set of all the possible values of the independent variable. Since the independent variable in the above equation is written in the form of a polynomial, we can say that the above equation is defined for all the real values of x.
Therefore, the domain of the given equation is $\left( -\infty ,\infty \right)$.
Now since y is a linear function of x, as can be seen in the above equation, the range will also be $\left( -\infty ,\infty \right)$.
Hence, the domain and the range are $\left( -\infty ,\infty \right)$ each.

Note: There is no information given in the above question regarding the dependent and the independent variables. Nor is any variable explicitly expressed in terms of the other. So we can choose any one of x and y as the independent variables. But we must note that this will not affect the domain and range of the given equation.