To determine whether the given system of linear equations has no solutions, infinitely many solutions, or exactly one solution, we can utilize the concept of determinants from linear algebra. Here's the given system of equations:
[tex]\[
\begin{aligned}
-6x + 5y &= -9 \\
-11x + 10y &= -17
\end{aligned}
\][/tex]
The corresponding coefficient matrix is:
[tex]\[
\begin{bmatrix}
-6 & 5 \\
-11 & 10
\end{bmatrix}
\][/tex]
The determinant (Δ) of the coefficient matrix is calculated as follows:
[tex]\[
\Delta = \text{det} \begin{bmatrix}
-6 & 5 \\
-11 & 10
\end{bmatrix} = (-6 \times 10) - (-11 \times 5) = -60 + 55 = -5
\][/tex]
The determinant is non-zero ([tex]\(\Delta \neq 0\)[/tex]). This implies that the system of equations is consistent and has exactly one unique solution.
Therefore, the given system of equations has exactly one solution.
The correct answer is:
One Solution
