Example 01:  Find the values of \(x\) and \(y\) if \(199 x + 201 y  = 600\) and \(201x  + 199y  = 200\)

Solution: At first look, coefficients of these equations seem very big numbers to handle. So multiplying equation (1) by 201 and equation (2) by 199 will not be a good idea to solve the equations. Since the coefficients of x and y are interchanged, we can add and subtract these equations, 

Adding the given equations, 

\(400x + 400y  = 800\) 

or  \(x + y = 2\) …(3) 

Subtracting equation (2) form equation (1), we get 

\(-2x + 2y = 400\) or 

\(- x  +  y  = 200\) …(4) 

Now equation (4) & equation (5) are very easy to solve by simple addition and subtraction.

\(y = 101\) and  \(x = -99\) 

Example 02: Find values of  \(x\) & \(y\) if \(254x + 246y  = 1032\) and \(246x + 254y  =  968\)

Solution: By adding equation (1) & equation (2) 

We get,

\(500x + 500y  = 2000\)  or  \(x + y = 4\) …(3) 

Subtracting second equation from first equation, 

We get,  \(8x  - 8y  = 64\) or 

\(x - y    =  8\)  …(4) 

Now by solving equation (3) & equation (4), we get,

\(x = 6, y = - 2\) 

\(x + y = 14,\;y + z = 17,\;z + x = 15\)

Adding all the three equations, we have

\(2(x + y + z) = 46\)

\( \Rightarrow x + y + z = 23\)

From the last equation, that we obtained, subtracting the first, second and third equations, we get the values of \(x = 6,y = 8,z = 9\)

Example: Find the values of \(x,y,z\) and \(w\) if
\(x + y + z - w = 17\), \(x + y - z + w = 9\), \(x - y + z + w = 19\) and \( - x + y + z + w = 13\)

Solution: The equations are written in a cyclic manner, adding all the equations, we have

\(2(x + y + z + w) = 58 \Rightarrow x + y + z + w = 29\)

From the last equation, subtract the four given equations, and we will get the values of \(w,z,y\) and \(x\) as 6, 10, 5 and 8 respectively.