https://file.aiquickdraw.com/mathbot/file/1727966432243b15det38.pdf

The third question on page 4 of the document is as follows:

Given the system of equations:

\[
\begin{cases}
x + (k + 1)y + 1 = 0 \\
2kx + 5y - 3 = 0 \\
3x + 7y + 1 = 0
\end{cases}
\]

Find the values of \( k \) for which the equations are consistent.

### Solution:

To check the consistency of the system, we follow the steps of determining the rank of the coefficient matrix and the augmented matrix, and checking if they are equal.

#### Step 1: Write the system in matrix form

The system of equations can be written in matrix form \( A \vec{x} = B \), where \( A \) is the coefficient matrix, \( \vec{x} \) is the vector of variables \( [x, y] \), and \( B \) is the vector of constants.

Coefficient matrix \( A \):

\[
A = \begin{bmatrix}
1 & (k + 1) \\
2k & 5 \\
3 & 7
\end{bmatrix}
\]

Constants matrix \( B \):

\[
B = \begin{bmatrix}
-1 \\
3 \\
-1
\end{bmatrix}
\]

#### Step 2: Find the augmented matrix

The augmented matrix is:

\[
[A|B] = \begin{bmatrix}
1 & (k + 1) & -1 \\
2k & 5 & 3 \\
3 & 7 & -1
\end{bmatrix}
\]

#### Step 3: Use elementary row operations and check for consistency

To determine consistency, we need to reduce the augmented matrix and check if the rank of the coefficient matrix \( A \) is equal to the rank of the augmented matrix \( [A|B] \).

After performing Gaussian elimination, you can solve for \( k \) and determine the values that ensure the system is consistent.

In conclusion, solving for the values of \( k \) that make the system consistent yields the answer:

\[
k = 2 \text{ or } k = -\frac{1}{2}.
\]

Let me know if you'd like further details on the solution process!

### Related Questions:

1. How is the rank of a matrix used to determine the consistency of a system of equations?
2. What is the significance of the determinant in analyzing the consistency of square systems?
3. How do free variables affect the solution space in consistent systems with infinite solutions?
4. What are the key steps involved in solving a system of equations using the rank method?
5. How do we determine if a system with zero determinant has multiple or no solutions?

**Tip:** When solving systems of equations, always check if the system is square (equal number of equations and unknowns) to simplify consistency checks using the determinant.

Learn how to determine the values of k for which a given system of equations is consistent. The problem involves matrix theory, rank of a matrix, and Gaussian elimination techniques, ideal for advanced students in Grades 11-12 or university level.

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