Question

The linear system of equations
𝑥 + 𝑦 = 3,
$𝑥 + (𝑘^2 − 8)𝑦 = 𝑘, 𝑘 ∈ 𝑅$
has no solution for 𝑘 = _____ (in integer).

Answer 1

Correct Answer: 3

Solution for Finding k Where the System Has No Solution 

Given Linear System of Equations:

\[ x + y = 3 \] \[ x + (k^2 - 8)y = k, \quad k \in \mathbb{R} \]

Step 1: Writing the Coefficient Matrix

The coefficient matrix of the system is:

\[ A = \begin{bmatrix} 1 & 1 \\ 1 & k^2 - 8 \end{bmatrix} \]

Step 2: Finding the Determinant

The determinant of matrix \( A \) is:

\[ \text{Det}(A) = (1)(k^2 - 8) - (1)(1) \]

Simplifying:

\[ \text{Det}(A) = k^2 - 9 \]

Step 3: Condition for No Solution

For the system to have no solution, the determinant must be zero:

\[ k^2 - 9 = 0 \]

Solving for \( k \):

\[ k^2 = 9 \]

\[ k = \pm 3 \]

Step 4: Finding the Integer Value

Since we need an integer value, we select:

\[ k = -3 \]

Final Answer:

The system has no solution for \( k = -3 \).