Question

5x − 7y + z = 11
6x − 8y − z = 15
3x + 2y − 6z = 7

Answer 1

Given: 5x − 7y + z = 11
6x − 8y − z = 15
3x + 2y − 6z = 7

$$\begin{gathered} D=\left|\begin{array}{ccc}5& -7& 1\\ 6& -8& -1\\ 3& 2& -6\end{array}\right| \\ =5(48+2)+7(-36+3)+1(12+24) \\ =5\left(50\right)+7(-33)+1\left(36\right) \\ =55 \\ {D}_1=\left|\begin{array}{ccc}11& -7& 1\\ 15& -8& -1\\ 7& 2& -6\end{array}\right| \\ =11(48+2)+7(-90+7)+1(30+56) \\ =11\left(50\right)+7(-83)+1\left(86\right) \\ =55 \\ {D}_2=\left|\begin{array}{ccc}5& 11& 1\\ 6& 15& -1\\ 3& 7& -6\end{array}\right| \\ =5(-90+7)-11(-36+3)+1(42-45) \\ =5(-83)-11(-33)+1(-3) \\ =-55 \\ {D}_3=\left|\begin{array}{ccc}5& -7& 11\\ 6& -8& 15\\ 3& 2& 7\end{array}\right| \\ =5(-56-30)+7(42-45)+11(12+24) \\ =5(-86)+7(-3)+11\left(36\right) \\ =-55 \\ \mathrm{Now}, \\ x=\frac{D_1}{D}=\frac{55}{55}=1 \\ y=\frac{D_2}{D}=\frac{-55}{55}=-1 \\ z=\frac{D_3}{D}=\frac{-55}{55}=-1 \\ \therefore x=1,y=-1\mathrm{and}z=-1 \end{gathered}$$