Question

The following determinants are obtained from the

simultaneous equations in variables x and y.

Dx = - 11 a

9 -4

Dy= 3 - 11

b 9

D= 3 2

7 -4

The solutions of the equation are x = - 1 and y = - 4. Find the values of a and b.

Also find original simultaneous equation having this solution.​

Answer 1

$\textbf{Given:}$

$\mathsf{D_x=\left|\begin{array}{cc}-11&a\\9&-4\end{array}\right|}$

$\mathsf{D_y=\left|\begin{array}{cc}3&-11\\b&9\end{array}\right|}$

$\mathsf{D=\left|\begin{array}{cc}3&2\\7&-4\end{array}\right|}$

$\textsf{Solution is x=-1 and y=-4}$

$\textbf{To find:}$

$\textsf{Values of a and b}$

$\textbf{Solution:}$

$\mathsf{D_x=\left|\begin{array}{cc}-11&a\\9&-4\end{array}\right|}$

$\mathsf{D_x=44-9a}$

$\mathsf{D_y=\left|\begin{array}{cc}3&-11\\b&9\end{array}\right|}$

$\mathsf{D_y=27+11b}$

$\mathsf{D=\left|\begin{array}{cc}3&2\\7&-4\end{array}\right|}$

$\mathsf{D=-12-14=-26}$

$\textsf{By Cramer's rule,}$

$\mathsf{x=\dfrac{D_x}{D}}$

$\implies\mathsf{-1=\dfrac{44-9a}{-26}}$

$\implies\mathsf{26=44-9a}$

$\implies\mathsf{26-44=-9a}$

$\implies\mathsf{-18=-9a}$

$\implies\mathsf{a=\dfrac{-18}{-9}}$

$\implies\boxed{\mathsf{a=2}}$

$\mathsf{y=\dfrac{D_y}{D}}$

$\implies\mathsf{-4=\dfrac{27+11b}{-26}}$

$\implies\mathsf{104=27+11b}$

$\implies\mathsf{77=11b}$

$\implies\mathsf{b=\dfrac{77}{11}}$

$\implies\boxed{\mathsf{b=7}}$

$\mathsf{Also,\;the\;equations\;are\;}$

$\mathsf{3x+2y=-11}$

$\mathsf{7x-4y=9}$

Answer 2

Answer:

Here is your answer

Step-by-step explanation: