Let's solve the system of linear equations for [tex]\(z\)[/tex] in detail. The system of equations is given by:
1. [tex]\(2x + 3y - 6z + 4w = 10\)[/tex]
2. [tex]\(8w - 3x - 4y + 6z = 7\)[/tex]
3. [tex]\(2y - 8z + 7x - w = -7\)[/tex]
4. [tex]\(6z + 4w - 5x - 7y = -21\)[/tex]
The objective is to solve this system specifically for the variable [tex]\(z\)[/tex]. Here's a step-by-step approach to solve the system:
Step 1: Express the equations in a more convenient form (standard form).
1. [tex]\(2x + 3y - 6z + 4w = 10\)[/tex]
2. [tex]\(-3x - 4y + 6z + 8w = 7\)[/tex]
3. [tex]\(7x + 2y - 8z - w = -7\)[/tex]
4. [tex]\(-5x - 7y + 6z + 4w = -21\)[/tex]
Step 2: Using elimination or substitution method, we solve the system of equations.
Step 3: Combine equations to eliminate variables step-by-step until only [tex]\(z\)[/tex] is isolated.
Given the complexity of a system involving multiple steps, we'll outline the critical stages and operations performed, which are usually done through matrix manipulation or processing linear combinations of these equations.
Step 4: Upon solving, we deduce the value of [tex]\(z\)[/tex].
From the algebraic manipulations and solving the equations systematically:
The value of [tex]\(z\)[/tex] is calculated to be:
[tex]\[ z = 3 \][/tex]
Thus, the value of [tex]\(z\)[/tex] in the system of equations is [tex]\( \boxed{3} \)[/tex].
