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To solve the system of linear equations provided in the image, we have three equations:

1. \( x - 4y + 4z = -1 \)
2. \( -x - z + 3y = 5 \)
3. \( 2x + 5z - 6 = y \)

We can rearrange the third equation to express \( y \) in terms of \( x \) and \( z \) and substitute into the first two equations. The third equation rearranged is:

\[ y = 2x + 5z - 6 \]

Substituting this into the first and second equations:

1. \( x - 4(2x + 5z - 6) + 4z = -1 \)
2. \( -x - z + 3(2x + 5z - 6) = 5 \)

Expanding and simplifying these equations:

1. \( x - 8x - 20z + 24 + 4z = -1 \)
2. \( -x - z + 6x + 15z - 18 = 5 \)

Further simplification:

1. \( -7x - 16z = -25 \)
2. \( 5x + 14z = 23 \)

Now, we have a simplified system:

1. \( -7x - 16z = -25 \)
2. \( 5x + 14z = 23 \)

We can solve these equations using substitution or elimination. Let's use the elimination method to find \( x \) and \( z \):

Multiplying the first equation by 5 and the second by 7 to align the coefficients of \( x \):

1. \( -35x - 80z = -125 \)
2. \( 35x + 98z = 161 \)

Adding these equations:

\[ 18z = 36 \]
\[ z = 2 \]

Now substitute \( z = 2 \) back into one of the simplified equations:

\[ 5x + 14(2) = 23 \]
\[ 5x + 28 = 23 \]
\[ 5x = -5 \]
\[ x = -1 \]

Now substituting \( x = -1 \) and \( z = 2 \) into the equation for \( y \):

\[ y = 2(-1) + 5(2) - 6 \]
\[ y = -2 + 10 - 6 \]
\[ y = 2 \]

Thus, the solution to the system is \( x = -1 \), \( y = 2 \), \( z = 2 \).

Would you like more details on any step of this solution or do you have any other questions? Here are some topics we could explore next:

1. How would the solution change if we used a different method, like matrix inversion or Gaussian elimination?
2. What would happen to the solution if we changed some of the coefficients in the equations?
3. What are the implications of having no solution or infinitely many solutions in a system of equations?
4. How can we verify the solution to ensure it's correct?
5. Would you like to practice solving another set of equations?

**Tip:** Always check the consistency of a system of linear equations. If during the process you derive a contradiction, such as \(0 = 1\), the system has no solution.

Solve the system of linear equations below: x - 4y + 4z = -1, -x - z + 3y = 5, 2x + 5z - 6 = y.

Discover how to solve a system of linear equations involving three variables using substitution and elimination methods. This detailed solution covers the steps needed to find the unique solution for x, y, and z. Suitable for high school students in Grades 10-12.

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