Let's solve the given system of linear equations step by step:
1. The system of equations is:
[tex]\[
\begin{cases}
2x + y + z = 92 \\
x + 2y + z = 8
\end{cases}
\][/tex]
2. Let's isolate one of the variables from one of the equations. For simplicity, we will work with the first equation to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[
2x + y + z = 92 \implies y = 92 - 2x - z
\][/tex]
3. Substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[
x + 2(92 - 2x - z) + z = 8
\][/tex]
Simplify and solve the resulting equation step by step:
[tex]\[
x + 184 - 4x - 2z + z = 8 \\
x - 4x + 184 + z - 2z = 8 \\
-3x + 184 - z = 8
\][/tex]
[tex]\[
-3x - z = -176 \\
z = -3x + 176
\][/tex]
4. Now, we have found expressions for [tex]\( y \)[/tex] and [tex]\( z \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 92 - 2x - z \\
y = 92 - 2x - (-3x + 176)
\][/tex]
Simplify the above:
[tex]\[
y = 92 - 2x + 3x - 176 \\
y = x - 84
\][/tex]
5. Therefore, the generalized solutions based on the discovered relationships are:
[tex]\[
x = x \\
y = x - 84 \\
z = 176 - 3x
\][/tex]
Based on this analysis, the solution set for the given system of equations can be written as:
[tex]\[
\left\{\begin{array}{l}
x = x \\
y = x - 84 \\
z = 176 - 3x
\end{array}\right.
\][/tex]
6. To find a specific solution, we can assume a value for one of the variables. Let's assume [tex]\( z = 0 \)[/tex].
[tex]\[
176 - 3x = 0 \implies 3x = 176 \implies x = \frac{176}{3}
\][/tex]
7. Substitute [tex]\( x = \frac{176}{3} \)[/tex] back into the equation for [tex]\( y \)[/tex]:
[tex]\[
y = x - 84 \implies y = \frac{176}{3} - 84
\][/tex]
[tex]\[
y = \frac{176}{3} - \frac{252}{3} = \frac{176 - 252}{3} = \frac{-76}{3}
\][/tex]
So, the specific solution set with [tex]\( z = 0 \)[/tex] is:
[tex]\[
\left\{\begin{array}{l}
x = \frac{176}{3} \\
y = \frac{-76}{3} \\
z = 0
\end{array}\right.
\][/tex]
Thus, the generalized solution to the system of equations is:
[tex]\[
x = \frac{176}{3} - \frac{z}{3}, \quad y = \frac{-76}{3} - \frac{z}{3}
\][/tex]
and a specific solution for [tex]\( z = 0 \)[/tex] is:
[tex]\[
x = \frac{176}{3}, \quad y = \frac{-76}{3}, \quad z = 0
\][/tex]
