Solve the simultaneous equations $4x+7y=6$ and $2x+5y=3$ using the matrix inverse method.

Step-by-step solution

1. Write the coefficient matrix $D=\left(\begin{array}{cc}4& 7\\ 2& 5\end{array}\right)$ and the constant vector $\mathbf{c}=\left(\begin{array}{c}6\\ 3\end{array}\right)$.
2. Compute the determinant: $\det D=4\cdot 5-7\cdot 2=20-14=6$.
3. Form the adjugate matrix: $\mathrm{adj}(D)=\left(\begin{array}{cc}5& -7\\ -2& 4\end{array}\right)$.
4. The inverse is $D^{-1}=\frac{1}{\det D}\mathrm{adj}(D)=\frac{1}{6}\left(\begin{array}{cc}5& -7\\ -2& 4\end{array}\right)$.
5. Multiply $D^{-1}$ with $\mathbf{c}$:
6. $x=\frac{1}{6}(5\cdot 6+(-7)\cdot 3)=\frac{1}{6}(30-21)=\frac{9}{6}=\frac{3}{2}$,
7. $y=\frac{1}{6}((-2)\cdot 6+4\cdot 3)=\frac{1}{6}(-12+12)=0$.
8. Hence the solution is $x=\frac{3}{2},y=0$.

Concept tested: Inverse of a 2×2 matrix to solve linear equations

Published 12 September 2026 · How this solution was prepared and checked