SAve usug inverse method \( \begin{array}{l}7 a+3 m-2 e=42 \\ a+m-e=11 \\ 6 a-m-9 e=18 \\ 10 a+2 m+0.5 e=83\end{array} \)

To tackle the given system of equations using the inverse method, we'll first express the equations in matrix form as \(AX = B\). Here, \(A\) is the matrix of coefficients, \(X\) contains the variables, and \(B\) is the constants on the right-hand side. The system can be represented as: \[ A = \begin{pmatrix} 7 & 3 & -2 \\ 1 & 1 & -1 \\ 6 & -1 & -9 \\ 10 & 2 & 0.5 \end{pmatrix}, \quad X = \begin{pmatrix} a \\ m \\ e \end{pmatrix}, \quad B = \begin{pmatrix} 42 \\ 11 \\ 18 \\ 83 \end{pmatrix} \] To find the solution \(X\), we compute \(X = A^{-1}B\). First, you need to calculate the inverse of matrix \(A\). Remember, matrix inverses can be tricky, so take your time ensuring all calculations are accurate. After finding \(A^{-1}\), multiply it by \(B\) to get the values of \(a\), \(m\), and \(e\). This will lead you directly to the solution of your system. Happy problem-solving! Additionally, ensure to double-check your calculations, as small mistakes in arithmetic can lead to completely off results! If you're ever in doubt, verifying your results by substituting them back into the original equations can help confirm that everything checks out perfectly.