To solve the system of equations:
[tex]\[ \begin{array}{c}
2x + 4y = 4 \\
-4x + 3y = -96
\end{array} \][/tex]
we can use the method of linear algebra where we transform the equations into matrix form. Here's the complete step-by-step solution:
1. Write the system of equations in matrix form [tex]\( Ax = B \)[/tex]:
[tex]\[ A = \begin{pmatrix} 2 & 4 \\ -4 & 3 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 4 \\ -96 \end{pmatrix} \][/tex]
2. Calculate the determinant of matrix [tex]\( A \)[/tex]:
The determinant [tex]\( \text{det}(A) \)[/tex] is calculated as:
[tex]\[ \text{det}(A) = 2 \cdot 3 - (-4) \cdot 4 = 6 + 16 = 22 \][/tex]
Since the determinant is non-zero (22 ≠ 0), the system of equations has a unique solution.
3. Find the solution [tex]\( x \)[/tex] using the inverse of matrix [tex]\( A \)[/tex] or a direct solver (like using Cramer's rule or matrix inversion methods):
Given that the determinant is non-zero, we use linear algebra techniques to find the unique solution for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Solving the equations simultaneously, we get:
[tex]\[
x = 18
\][/tex]
[tex]\[
y = -8
\][/tex]
4. Conclusion:
The system of equations has exactly one solution, which is [tex]\( x = 18 \)[/tex] and [tex]\( y = -8 \)[/tex].
Therefore, the system has one solution. The solution set is:
[tex]\[ \{ (18, -8) \} \][/tex].
