Question

If the system of equations has a unique solution, find the values of \( a \) and \( b \).

• A. \( a = 8, b = 15 \)
• B. \( a \neq 8, b \in \mathbb{R} \)
• C. \( a = 8, b \neq 15 \)
• D. \( a \neq 15, b = 8 \)

Hint:

For a unique solution in a system of equations, ensure the determinant of the coefficient matrix is non-zero.

Answer 1

The Correct Option is B

Step 1: Constructing the Coefficient Matrix The given system of equations corresponds to the coefficient matrix: \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 5 & a\\ \end{bmatrix} \] For the system to have a unique solution, the determinant of the matrix \( A \) must be non-zero, i.e., \( |A| \neq 0 \).
Step 2: Computing the Determinant Expanding along the first row: \[ |A| = \begin{vmatrix} 1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 5 & a\\ \end{vmatrix} \] \[ = 1 \begin{vmatrix} 3 & 5 \\ 5 & a\\ \end{vmatrix} - 2 \begin{vmatrix} 1 & 5 \\ 2 & a\\ \end{vmatrix} + 3 \begin{vmatrix} 1 & 3 \\ \end{vmatrix} \] Calculating the minors: \[ \begin{vmatrix} 3 & 5 \\ 5 & a\\ \end{vmatrix} = (3a - 25) \] \[ \begin{vmatrix} 1 & 5 \\ 2 & a\\ \end{vmatrix} = (a - 10) \] \[ \begin{vmatrix} 1 & 3 \\ 2 & 5 \\ \end{vmatrix} = (5 - 6) = -1 \] Substituting these: \[ |A| = 1(3a - 25) - 2(a - 10) + 3(-1) \] \[ = 3a - 25 - 2a + 20 - 3 = a - 8 \]
Step 3: Condition for Unique Solution For a unique solution: \[ a - 8 \neq 0 \quad \Rightarrow \quad a \neq 8. \]
Step 4: Determining \( b \) Since the determinant only depends on \( a \), \( b \) can take any real value: \[ b \in \mathbb{R}. \] Thus, the correct answer is: \[ \boxed{a \neq 8, b \in \mathbb{R}} \]