Answer: The required solution of the given system is
x = 3, y = 6 and z = -1.
Step-by-step explanation: We are given to solve the following system of equations by the method of substitution :
[tex]-x-y-z=-8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\-4x+4y+5z=7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)\\\\2x+2z=4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)[/tex]
From equation (iii), we have
[tex]2x+2z=4\\\\\Rightarrow x+z=2\\\\\Rightarrow x=2-z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iv)[/tex]
Substituting the value of x from equation (iv) in equations (i) and (ii), we get
[tex]-(2-z)-y-z=-8\\\\\Rightarrow 2-z+y+z=8\\\\\Rightarrow 2+y=8\\\\\Rightarrow y=8-2\\\\\Rightarrow y=6[/tex]
and
[tex]-4(2-z)+4y+5z=7\\\\\Rightarrow -8+4z+4\times6+5z=7\\\\\Rightarrow 9z+16=7\\\\\Rightarrow 9z=-9\\\\\Rightarrow z=-\dfrac{9}{9}\\\\\Rightarrow z=-1.[/tex]
From equation (iii), we get
[tex]2x+2(-1)=4\\\\\Rightarrow 2x=6\\\\\Rightarrow x=\dfrac{6}{2}\\\\\Rightarrow x=3.[/tex]
Thus, the required solution of the given system is
x = 3, y = 6 and z = -1.