Solve the system $3x-y=7$, $2x+4y=-2$ using the matrix inverse method. Choose the correct ordered pair $(x,y)$.

Step-by-step solution

1. Write the coefficient matrix $C=\left(\begin{array}{cc}3& -1\\ 2& 4\end{array}\right)$ and the constant vector $\mathbf{b}=\left(\begin{array}{c}7\\ -2\end{array}\right)$.
2. Find the determinant of $C$: $\det C=3\cdot 4-(-1)\cdot 2=12+2=14$.
3. Since $\det C\ne 0$, $C$ is invertible. Compute its inverse:
4. $C^{-1}=\frac{1}{14}\left(\begin{array}{cc}4& 1\\ -2& 3\end{array}\right)$ (swap diagonal entries and change sign of off‑diagonal).
5. Multiply the inverse with $\mathbf{b}$: $\mathbf{x}=C^{-1}\mathbf{b}=\frac{1}{14}\left(\begin{array}{cc}4& 1\\ -2& 3\end{array}\right)\left(\begin{array}{c}7\\ -2\end{array}\right)$.
6. Calculate $x=\frac{1}{14}(4\cdot 7+1\cdot (-2))=\frac{26}{14}=\frac{13}{7}$.
7. Calculate $y=\frac{1}{14}(-2\cdot 7+3\cdot (-2))=\frac{-20}{14}=\frac{-10}{7}$.

Concept tested: Inverse of a $2\times2$ matrix to solve linear equations

Published 15 September 2026 · How this solution was prepared and checked