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: \[ \boxed{a \neq 8, b \in \mathbb{R}} \]

Answer 2

To determine the conditions for a unique solution of a system of equations, consider the general form of two linear equations:

\( ax + by = c \)

\( dx + ey = f \)

For the system to have a unique solution, the determinant of the coefficient matrix must be non-zero. The determinant for a 2x2 system is given by:

\( \text{Determinant} = ae - bd \)

To find the unique solution condition, this determinant should not equal zero:

\( ae - bd \neq 0 \)

Assuming we have a specific system with:

\( a = 8 \)

\( e = b \)

We require:

\( 8e - (bd) \neq 0 \)

If \( a = 8 \), \( b \neq 0 \) must hold, but to satisfy all options:
The above options indicate that the determinant reduces correctly to non-zero for all \( b \in \mathbb{R} \), unless \( a \neq 8 \). By examining the structure of the options, the condition \( a \neq 8 \) makes the system consistent with all \( b \).

Thus, the condition for a unique solution is:

\( a \neq 8, b \in \mathbb{R} \)