To determine whether the given system of equations has a unique solution, we need to examine the determinant of the coefficient matrix. Let's break down the steps:
1. Write down the system of equations:
[tex]\[
\begin{cases}
6x - 8y = 5 \\
-3x + 4y = -4
\end{cases}
\][/tex]
2. Form the coefficient matrix:
The coefficients of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the equations form the matrix:
[tex]\[
\begin{bmatrix}
6 & -8 \\
-3 & 4
\end{bmatrix}
\][/tex]
3. Calculate the determinant of the coefficient matrix:
To find the determinant of the matrix
[tex]\[
\begin{bmatrix}
6 & -8 \\
-3 & 4
\end{bmatrix}
\][/tex]
we use the formula for the determinant of a 2x2 matrix:
[tex]\[
\text{det}(A) = ad - bc
\][/tex]
where [tex]\(a = 6\)[/tex], [tex]\(b = -8\)[/tex], [tex]\(c = -3\)[/tex], and [tex]\(d = 4\)[/tex]. Now substitute these values into the formula:
[tex]\[
\text{det}(A) = (6 \cdot 4) - (-8 \cdot -3) = 24 - 24 = 0
\][/tex]
4. Interpret the determinant:
The determinant of the coefficient matrix is [tex]\(0\)[/tex].
5. Conclusion about the solution:
If the determinant of the coefficient matrix is [tex]\(0\)[/tex], it means that the system of equations does not have a unique solution. In this case, the system could either have no solution or infinitely many solutions, but it does not have exactly one unique solution.
Thus, the correct answer is:
```
No, because the determinant of the coefficient matrix is 0.
```
