The problem provided is a system of linear equations, and we need to find the value of $x^2 - y^2 + z^2$. Here’s a breakdown:

Given System of Equations:

1. $x + 2y + 4z = 9$
2. $2x - 2y + 6z = 26$
3. $3x + 4y - 8z = -29$

Steps to Solve:

1. Solve the system of equations to find the values of $x$, $y$, and $z$.
2. Substitute these values into the expression $x^2 - y^2 + z^2$ to find its value.

I will calculate the values of $x$, $y$, and $z$ and then determine the value of $x^2 - y^2 + z^2$.The solution to the system of equations is:

$x = \frac{45}{29}, \quad y = -\frac{68}{29}, \quad z = \frac{88}{29}$

Now, we need to find the value of $x^2 - y^2 + z^2$. I'll calculate that next.The value of $x^2 - y^2 + z^2$ is:

$x^2 - y^2 + z^2 = \frac{5145}{841} = 6.12$

Rounding to the nearest integer, the answer is approximately 6, which corresponds to option D.

Would you like further details or have any other questions? Here are 5 related questions for deeper understanding:

1. How do you solve a system of linear equations using matrix methods?
2. What are some common applications of solving linear equations in real life?
3. How can we interpret the geometrical meaning of solving three linear equations?
4. What are some strategies for checking the consistency of a system of equations?
5. What is the significance of determinants in solving systems of equations?

Tip: When dealing with fractions in equations, it's helpful to use exact values during intermediate steps to avoid rounding errors.