Questions: Solve the system using inverse matrices. 5x - 2y = 30 x + 6y = -26

Solution Steps

To solve the system of equations using inverse matrices, we first represent the system in matrix form \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the column matrix of variables, and \( B \) is the column matrix of constants. We then find the inverse of matrix \( A \) and multiply it by matrix \( B \) to find matrix \( X \), which contains the solutions for \( x \) and \( y \).

We start with the system of equations given by: \[ \begin{align_} 5x - 2y &= 30 \quad (1) \\ x + 6y &= -26 \quad (2) \end{align_} \] This can be represented in matrix form as \( AX = B \), where: \[ A = \begin{bmatrix} 5 & -2 \\ 1 & 6 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \end{bmatrix}, \quad B = \begin{bmatrix} 30 \\ -26 \end{bmatrix} \]

To solve for \( X \), we first need to find the inverse of matrix \( A \): \[ A^{-1} = \begin{bmatrix} 0.1875 & 0.0625 \\ -0.03125 & 0.15625 \end{bmatrix} \]

Next, we multiply the inverse of \( A \) by \( B \) to find \( X \): \[ X = A^{-1}B = \begin{bmatrix} 0.1875 & 0.0625 \\ -0.03125 & 0.15625 \end{bmatrix} \begin{bmatrix} 30 \\ -26 \end{bmatrix} = \begin{bmatrix} 4 \\ -5 \end{bmatrix} \]

Final Answer