Theory:

The elimination method is another method of solving the system of linear equations.
 
The steps to solve the system of linear equations is given by:
Step 1: First, multiply either one equation or both the equations by a number such that in both the equations, either the coefficients of the first variable or the coefficients of the second variable are equal.
 
Step 2: Add both the equations or subtract one equation from the other so that the equal coefficients gets cancelled.
 
Step 3: Solve the obtained equation in Step 2 to find the value of one of the variables.
 
Step4: Substitute the value of one of the variables(obtained in Step 3) in any one of the two equations to find the value of the remaining variable.
Let us consider an example to understand the concept clearly.
Example:
Solve the simultaneous linear equations \(2x+y = 8\) and \(x+2y = 10\) by elimination method.
 
Solution:
 
\(2x+y = 8\) ---- (\(1\))
 
\(x+2y = 10\) ---- (\(2\))
 
Step 1: Let us multiply equation (\(1\)) by number \(2\).
 
Thus, we have:
 
\(4x+2y =16\)
 
Step 2: Now, subtract equation (\(2\)) from equation (\(1\)) and cancel the equal coefficients of \(y\).
 
4x+2y=16x+2y=10¯3x+0y=6
 
Step 3: Solve the equation to find the value of the variable \(x\).
 
\(3x = 6\)
 
\(x = 2\)
 
Thus, the value of the variable \(x\) is \(2\).
 
Step 4: Substitute the value of \(x\) in equation (\(2\)).
 
Thus, we have:
 
\(2+2y = 10\)
 
\(2y = 10-2\)
 
\(2y = 8\)
 
\(y = 4\)
 
Thus, the value of the variable \(y\) is \(4\).
 
Therefore, the solution of the given system of equations is \(x = 2\) and \(y = 4\).
Important!
1. In any of the steps, if we get a false equation like \(0 = 1\), then the system has no solution.
 
2. If we get an equation like \(0 = 0\), then the system has infinitely many solutions.