Consider the following system of linear equations: (2+ ^2 )x + ( ^2 )y + ( ^2 )z = 1 ( ^2 )x + (2+ ^2 )y + ( ^2 )z = 4 ( 2 )x + ( 2 )y + (2+ 2 )z = 7 The maximum value of x + y + z as varies over all real numbers is

Step-by-step solution

Let the given system of equations be: (2+ ^2 )x + ( ^2 )y + ( ^2 )z = 1 ... (i) ( ^2 )x + (2+ ^2 )y + ( ^2 )z = 4 ... (ii) ( 2 )x + ( 2 )y + (2+ 2 )z = 7 ... (iii) Adding equations (i), (ii), and (iii) vertically, we sum the coefficients of x , y , and z respectively: Coefficient of x : (2+ ^2 ) + ^2 + 2 = 2 + 1 + 2 = 3 + 2 Coefficient of y : ^2 + (2+ ^2 ) + 2 = 3 + 2 Coefficient of z : ^2 + ^2 + (2+ 2 ) = 3 + 2 Thus, the sum of the three equations yields: (3 + 2 )x + (3 + 2 )y + (3 + 2 )z = 1 + 4 + 7 (3 + 2 )(x +