It seems you're working with a system of linear equations involving three variables $x$, $y$, and $z$. Let's rewrite and solve this system.

The system of equations is:

1. $4x + y + z = -10$
2. $5x - 3y - z = -7$
3. $6x + y - z = 8$

Step 1: Add or subtract the equations to eliminate one variable.

• We will first attempt to eliminate $z$.

Let's add equations (1) and (3) to eliminate $z$: $(4x + y + z) + (6x + y - z) = -10 + 8$ $10x + 2y = -2 \quad \text{(Equation A)}$

Now, let's add equations (2) and (3) to eliminate $z$: $(5x - 3y - z) + (6x + y - z) = -7 + 8$ $11x - 2y = 1 \quad \text{(Equation B)}$

Step 2: Solve the new system of two equations.

You now have the system:

1. $10x + 2y = -2$
2. $11x - 2y = 1$

Add these two equations to eliminate $y$: $(10x + 2y) + (11x - 2y) = -2 + 1$ $21x = -1$ $x = -\frac{1}{21}$

Step 3: Substitute $x = -\frac{1}{21}$ into one of the equations to find $y$.

Substitute into $10x + 2y = -2$: $10\left(-\frac{1}{21}\right) + 2y = -2$ $-\frac{10}{21} + 2y = -2$ Multiply everything by 21 to clear the fractions: $-10 + 42y = -42$ $42y = -32$ $y = -\frac{32}{42} = -\frac{16}{21}$

Step 4: Substitute $x = -\frac{1}{21}$ and $y = -\frac{16}{21}$ into one of the original equations to find $z$.

Substitute into the first equation $4x + y + z = -10$: $4\left(-\frac{1}{21}\right) + \left(-\frac{16}{21}\right) + z = -10$ $-\frac{4}{21} - \frac{16}{21} + z = -10$ $-\frac{20}{21} + z = -10$ Multiply by 21: $-20 + 21z = -210$ $21z = -190$ $z = -\frac{190}{21}$

Solution:

The solution to the system of equations is: $x = -\frac{1}{21}, \quad y = -\frac{16}{21}, \quad z = -\frac{190}{21}$

Would you like more details or clarifications?

Related Questions:

1. How do you use substitution to solve systems of equations?
2. What are the advantages of the elimination method for solving equations?
3. Can systems of linear equations have no solutions or infinitely many solutions?
4. What are some real-world applications of systems of linear equations?
5. How can you verify the solution to a system of equations?

Tip: When solving a system of equations, check if the variables cancel out efficiently by adding or subtracting equations.