Question

The students of a class are made to stand in rows. If 1 student is extra in a row, there would be 2 rows less. If 1 student is less in a row there would be 3 rows more. Find the number of students in the class.

Answer 1

Hint: For solving the above question, we would be requiring the knowledge of solving the system of linear equations in two variables. In this question we would be using an elimination method.

In elimination method, we first try to make the coefficient of any one variable of the two as equal and then subtract or add the new equations accordingly.
Then, we will get the equation which will have only one variable.

Then we can solve the equation to get the value of that variable which is left and after getting the value of any one variable, we can plug in that value in any of the equations and then get the value of the other variable as well.

One more important thing is that the total number of students in the class will remain the same and the product of the number of students in each row and the total number of rows gives us the number of the students in the class.

Complete step-by-step answer:
As mentioned in the question,
Let the number of students in each row be y and the number of rows be x.
Now, according to the question, we get
\[\begin{align}
  & (x-2)(y+1)=xy \\
 & xy+x-2y-2=xy \\
 & x-2y=2\ \ \ \ \ ...(a) \\
\end{align}\]
Now, again using the information that is given in the question, we get
\[\begin{align}
  & (x+3)(y-1)=xy \\
 & xy-x+3y-3=xy \\
 & -x+3y=3\ \ \ \ \ ...(b) \\
\end{align}\]

Now, on adding equations (a) and (b), we get
\[\begin{align}
  & \ \ \ \ x-2y=2 \\
 & \dfrac{+(-x+3y=3)}{\ \ \ \ \ \ \ \ \ \ \ \ y=5} \\
 & y=5 \\
\end{align}\]

Now, putting this value in (a), we get
\[\begin{align}
  & x-2\times 5=2 \\
 & x=12 \\
\end{align}\]
Hence, the number of students in each row is 5 and the number of rows is 12.
Hence, the total number of the students in the class is 60.

Note: For questions in which there are more than 2 variables, in order to know whether the equations are solvable or whether we will be able to get the values of the variables by just counting the number of variables and number of the equations. If the number of equations and the number of variables involved in the question is equal then we can surely say that every variable will be having a unique value. If these numbers are not equal, then we do not comment on that.
There are two other methods of solving a 2 variable system of equations:-
1. substitution method
2. cross multiplication method