Question

Find two solutions of the equation $3x - 2y = 5$.

Answer 1

Hint: The given equation is a linear equation in two variables. This will have infinitely many solutions because to find a unique solution we must have at least two equations. So put any two random values of $x$ and find the corresponding values of $y$ to obtain two solutions of the equation.

Complete step-by-step solution:
According to the question, we have been given an equation and we have to find two solutions to it.
The given equation is:
$ \Rightarrow 3x - 2y = 5$
As we can see that this is a linear equation in two variables $x$ and $y$. We know that to find a unique solution for such a system, we must have at least two equations. But this is only one equation, so this will give us infinitely many solutions.
We will put any two random values of $x$ and determine the corresponding value of $y$ to obtain two solutions of the equation.
If we put $x = 0$, we have:
$
   \Rightarrow 3\left( 0 \right) - 2y = 5 \\
   \Rightarrow - 2y = 5 \\
   \Rightarrow y = - \dfrac{5}{2} \\
 $
So one of its solutions is $\left( {0, - \dfrac{5}{2}} \right)$
Next, if we put $x = 1$, we have:
$
   \Rightarrow 3\left( 1 \right) - 2y = 5 \\
   \Rightarrow 3 - 2y = 5 \\
   \Rightarrow - 2y = 2 \\
   \Rightarrow y = - 1 \\
 $
Other solution of the equation is $\left( {1, - 1} \right)$

Thus two solutions of the equation $3x - 2y = 5$ are $\left( {0, - \dfrac{5}{2}} \right)$ and $\left( {1, - 1} \right)$.

Note: If a linear equation is having only one variable, it can be solved directly to get the value of the variable. If it is a two variable equation, we can’t find a unique solution from one equation. To determine the values of two different variables, we need a system of two different equations in those variables and to determine three variables we need a system of three equations in those variables. Similarly if there are $n$ different variables then we require $n$ different equations to find their values.