Question

Find the number of solutions for the equation using Cramer's rule:

$5x-7y=-27$ and $x-2y=-9$

• A. unique solution
• B. many solution
• C. no solution
• D. two solution

Answer 1

The correct option is A unique solution

Given equations are $5x-7y=-27,x-2y=-9$

Using Cramer's rule, find the determinant of the coefficient matrix,

$D=\left|\begin{array}{cc}5& -7\\ 1& -2\end{array}\right|=5\times -2-(1\times -7)$

$D=-10+7=-3$

Secondly, find the determinant of $x$ coefficient matrix,

$D_x=\left|\begin{array}{cc}-27& -7\\ -9& -2\end{array}\right|=-27\times -2-(-9\times -7)$

$D_x=5463=-9$

Similarly, find the determinant of $y$ coefficient matrix,

$D_y=\left|\begin{array}{cc}5& -27\\ 1& -9\end{array}\right|=5\times -9-(1\times -27)$

$\therefore D_y=-45+27=-18$

Therefore, $D=-3\ne 0,D_x=-9\ne 0,D_y=-18\ne 0$ the equation has an unique solution.