Solve the following simultaneous equations by using Cramers’s rule: 3x-y =7, x +4y =11

Answer: \(x = 3,\ y = 2\)

Equations: \(3x - y = 7\) and \(x + 4y = 11\).

\[D = \begin{vmatrix}3 & -1\\ 1 & 4\end{vmatrix} = 12 + 1 = 13\]

\[D_x = \begin{vmatrix}7 & -1\\ 11 & 4\end{vmatrix} = 28 + 11 = 39\]

\[D_y = \begin{vmatrix}3 & 7\\ 1 & 11\end{vmatrix} = 33 - 7 = 26\]

By Cramer's rule, \(x = \dfrac{D_x}{D} = \dfrac{39}{13} = 3\) and \(y = \dfrac{D_y}{D} = \dfrac{26}{13} = 2\).

Check: \(3(3) - 2 = 7\) and \(3 + 4(2) = 11\).