x+y+z=6.
x-y+2z=5.
x-y-3z=-10. Ley gausiana hacia atras 

To solve the system of equations using Gaussian elimination (also known as row reduction) with backward substitution, follow these steps:

1. \( x + y + z = 6 \)
2. \( x - y + 2z = 5 \)
3. \( x - y - 3z = -10 \)

First, write the augmented matrix for the system:
\[
\begin{pmatrix}
1 & 1 & 1 & | & 6 \\
1 & -1 & 2 & | & 5 \\
1 & -1 & -3 & | & -10
\end{pmatrix}
\]

Next, perform row operations to transform this matrix into an upper triangular form.

Step 1: Subtract the first row from the second and third rows to eliminate the \( x \) term from those rows.

\[
R2 = R2 - R1
\]
\[
\begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -2 & 1 & | & -1 \\
1 & -1 & -3 & | & -10
\end{pmatrix}
\]

\[
R3 = R3 - R1
\]
\[
\begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -2 & 1 & | & -1 \\
0 & -2 & -4 & | & -16
\end{pmatrix}
\]

Step 2: Subtract the second row from the third row to eliminate the \( y \) term from the third row.

\[
R3 = R3 - R2
\]
\[
\begin{pmatrix}
1 & 1 & 1 & | & 6 \\
0 & -2 & 1 & | & -1 \\
0 & 0 & -5 & | & -15
\end{pmatrix}
\]

Now we have an upper triangular matrix. We can start backward substitution to find the values of \( x \), \( y \), and \( z \).

From the third row:
\[
-5z = -15
\]
\[
z = 3
\]

From the second row:
\[
-2y + z = -1
\]
Substitute \( z = 3 \):
\[
-2y + 3 = -1
\]
\[
-2y = -4
\]
\[
y = 2
\]

From the first row:
\[
x + y + z = 6
\]
Substitute \( y = 2 \) and \( z = 3 \):
\[
x + 2 + 3 = 6
\]
\[
x + 5 = 6
\]
\[
x = 1
\]

So, the solution to the system is:
\[
x = 1
\]
\[
y = 2
\]
\[
z = 3
\]

Would you like more details or have any questions?

Here are 5 questions you might want to ask:
1. Can you explain the steps of Gaussian elimination in more detail?
2. How does Gaussian elimination differ from other methods of solving systems of equations?
3. Can you show an example of solving a system with more variables using Gaussian elimination?
4. What are the potential pitfalls or common mistakes in using Gaussian elimination?
5. How can we determine if a system of equations has no solution or infinitely many solutions using Gaussian elimination?

**Tip:** When using Gaussian elimination, always ensure to keep track of row operations clearly to avoid calculation errors and achieve an accurate upper triangular matrix.

Learn how to solve the system of equations x + y + z = 6, x - y + 2z = 5, x - y - 3z = -10 using Gaussian elimination with backward substitution. This problem covers advanced topics in linear algebra and systems of equations. Suitable for advanced high school students and beyond.

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