test the consistency and solve x+y+z=3 x+2y+3z=9 x+4y+9z=6

To test the consistency of the given system of equations and solve for \(x\), \(y\), and \(z\), we start by writing the system as:

\[
x + y + z = 3
\]
\[
x + 2y + 3z = 9
\]
\[
x + 4y + 9z = 6
\]

### Step 1: Form the augmented matrix
The augmented matrix for this system is:
\[
\begin{bmatrix}
1 & 1 & 1 & | & 3 \\
1 & 2 & 3 & | & 9 \\
1 & 4 & 9 & | & 6 \\
\end{bmatrix}
\]

### Step 2: Perform row reduction to echelon form

1. **Row 1:** \([1 \,\, 1 \,\, 1 \,\, | \,\, 3]\)
   
2. **Row 2:** Subtract Row 1 from Row 2:
   \[
   (1 - 1) \,\, (2 - 1) \,\, (3 - 1) \,\, | \,\, (9 - 3)
   \]
   Resulting in Row 2 as:
   \([0 \,\, 1 \,\, 2 \,\, | \,\, 6]\)
   
3. **Row 3:** Subtract Row 1 from Row 3:
   \[
   (1 - 1) \,\, (4 - 1) \,\, (9 - 1) \,\, | \,\, (6 - 3)
   \]
   Resulting in Row 3 as:
   \([0 \,\, 3 \,\, 8 \,\, | \,\, 3]\)

Now, the matrix is:
\[
\begin{bmatrix}
1 & 1 & 1 & | & 3 \\
0 & 1 & 2 & | & 6 \\
0 & 3 & 8 & | & 3 \\
\end{bmatrix}
\]

4. **Row 3:** Subtract 3 times Row 2 from Row 3:
   \[
   (3 - 3 \times 1) \,\, (8 - 3 \times 2) \,\, | \,\, (3 - 3 \times 6)
   \]
   Resulting in Row 3 as:
   \([0 \,\, 0 \,\, 2 \,\, | \,\, -15]\)

The matrix now is:
\[
\begin{bmatrix}
1 & 1 & 1 & | & 3 \\
0 & 1 & 2 & | & 6 \\
0 & 0 & 2 & | & -15 \\
\end{bmatrix}
\]

5. **Row 3:** Divide by 2 to make the last pivot equal to 1:
   \([0 \,\, 0 \,\, 1 \,\, | \,\, -7.5]\)

Now, the matrix is:
\[
\begin{bmatrix}
1 & 1 & 1 & | & 3 \\
0 & 1 & 2 & | & 6 \\
0 & 0 & 1 & | & -7.5 \\
\end{bmatrix}
\]

6. **Row 2:** Subtract 2 times Row 3 from Row 2:
   \((2 - 2 \times 1) \,\, | \,\, (6 - 2 \times -7.5)\)
   Resulting in Row 2 as:
   \([0 \,\, 1 \,\, 0 \,\, | \,\, 21]\)

7. **Row 1:** Subtract Row 3 from Row 1:
   \((1 - 1) \,\, | \,\, (3 - -7.5)\)
   Resulting in Row 1 as:
   \([1 \,\, 1 \,\, 0 \,\, | \,\, 10.5]\)

8. **Row 1:** Subtract Row 2 from Row 1:
   Resulting in Row 1 as:
   \([1 \,\, 0 \,\, 0 \,\, | \,\, -10.5]\)

The final row-reduced matrix is:
\[
\begin{bmatrix}
1 & 0 & 0 & | & -10.5 \\
0 & 1 & 0 & | & 21 \\
0 & 0 & 1 & | & -7.5 \\
\end{bmatrix}
\]

### Solution
The solutions are:
\[
x = -10.5, \quad y = 21, \quad z = -7.5
\]

### Consistency
Since we obtained a unique solution without encountering any contradictions, the system of equations is consistent.

Would you like further details or have any questions?

Here are 5 related questions:
1. How can we check if a system has infinitely many solutions?
2. What is the significance of an augmented matrix in solving linear equations?
3. How does row reduction simplify solving linear systems?
4. Can matrix methods be used to solve nonlinear equations?
5. What are the conditions for a matrix to be invertible?

**Tip:** Always check the determinant of the coefficient matrix when solving linear systems; if it's zero, the system might be inconsistent or have infinitely many solutions.

test the consistency and solve x+y+z=3, x+2y+3z=9, x+4y+9z=6

Solve the system of linear equations x + y + z = 3, x + 2y + 3z = 9, and x + 4y + 9z = 6 using Gaussian elimination. This detailed solution explores augmented matrices and row reduction techniques to determine if the system is consistent. Suitable for Grades 10-12.

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