Certainly! Let's analyze the given system of linear equations and determine if it has a unique solution.
The system is:
[tex]\[
\begin{cases}
2x - 3y = 5 \\
-4x + 6y = -4
\end{cases}
\][/tex]
To determine if this system has a unique solution, we need to examine the determinant of the coefficient matrix of the system.
First, let's write the coefficient matrix:
[tex]\[
\begin{pmatrix}
2 & -3 \\
-4 & 6
\end{pmatrix}
\][/tex]
The determinant of a 2x2 matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex] is calculated as [tex]\(ad - bc\)[/tex].
For our coefficient matrix [tex]\(\begin{pmatrix} 2 & -3 \\ -4 & 6 \end{pmatrix}\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -4\)[/tex]
- [tex]\(d = 6\)[/tex]
Thus, the determinant is:
[tex]\[
\text{det} = (2 \cdot 6) - (-4 \cdot -3) = 12 - 12 = 0
\][/tex]
We see that the determinant of the coefficient matrix is 0.
When the determinant of the coefficient matrix of a system of linear equations is zero, it implies that the system does not have a unique solution. The system might either have no solution (inconsistent) or have infinitely many solutions (dependent), but it definitely does not have a unique solution.
Thus, the correct answer is:
No, because the determinant of the coefficient matrix is 0.
