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For what values of a the system of equation is consistent

To determine the values of \( a \) for which the given system of equations is consistent, we must analyze the following system:

\[
\begin{cases}
x - y + z = 1 \\
x + 2y + 4z = a \\
x + 4y + 6z = a_2
\end{cases}
\]

### Step 1: Write the System in Matrix Form

The augmented matrix for this system is:

\[
\begin{bmatrix}
1 & -1 & 1 & | & 1 \\
1 & 2 & 4 & | & a \\
1 & 4 & 6 & | & a_2 \\
\end{bmatrix}
\]

### Step 2: Perform Row Operations

1. **Subtract Row 1 from Row 2 and Row 3** to make the first column zeroes below the first row.

   This leads to:
   \[
   \begin{bmatrix}
   1 & -1 & 1 & | & 1 \\
   0 & 3 & 3 & | & a - 1 \\
   0 & 5 & 5 & | & a_2 - 1 \\
   \end{bmatrix}
   \]

2. **Subtract Row 2 from Row 3** to simplify further:

   \[
   \begin{bmatrix}
   1 & -1 & 1 & | & 1 \\
   0 & 3 & 3 & | & a - 1 \\
   0 & 0 & 0 & | & a_2 - a \\
   \end{bmatrix}
   \]

### Step 3: Consistency Condition

For the system to be consistent, the last row \([0, 0, 0 | a_2 - a]\) implies that the system will have a solution if:
\[
a_2 - a = 0 \Rightarrow a_2 = a
\]

### Conclusion

The system is consistent if \( a_2 = a \).

Would you like more details on these steps, or do you have any specific questions?

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### Related Questions

1. How can we determine if a system has infinitely many solutions?
2. What does it mean for a system of equations to be inconsistent?
3. How can we solve for \( x \), \( y \), and \( z \) when \( a = a_2 \)?
4. What other methods can be used to check the consistency of a system?
5. How would the solution change if the equations were non-linear?

### Tip

In general, to check the consistency of a system, focus on the row echelon form to see if any contradictions (like a row of zeroes equaling a non-zero constant) arise.

For what values of a the system of equations is consistent?

Explore how to determine the values of parameter 'a' for which a system of linear equations remains consistent. This problem involves matrix row reduction and the conditions for a consistent system, suitable for students in Grades 11-12 or early college.

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