Does the following system have a unique solution? Why?

[tex]\[
\left\{\begin{array}{l}
2x - 3y = 5 \\
-4x + 6y = -4
\end{array}\right.
\][/tex]

A. No, because the determinant of the coefficient matrix is 0.
B. No, because the determinant of the coefficient matrix is 12.
C. Yes, because the determinant of the coefficient matrix is 0.
D. Yes, because the determinant of the coefficient matrix is 12.



Answer :

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.