Question

FURTHER MATHEMATICS Use determinants to solve the systems of equation:

2x + y + 2z = 13
X + y - 2z = 8
X + 2y + z = 11​

Answer 1

Explanation:

`\left\{\begin{array}{ccc}2x+1y+2z&=&13\\1x+1y-2z&=&8\\1x+2y+1z&=&11\\\end{array}\right.\\\\\\\Delta=\left| \begin{array}{ccc}2&1&2\\1&1&-2\\1&2&1\end{array}\right| =2*\left| \begin{array}{ccc}1&(1)/(2)&1\\1&1&-2\\1&2&1\end{array}\right| =2*\left| \begin{array}{ccc}1&(1)/(2)&1\\0&(1)/(2)&-3\\0&1&3\end{array}\right| =2*((3)/(2)+3)=9\\\\`

`\Delta_1=\left| \begin{array}{ccc}13&1&2\\8&1&-2\\11&2&1\end{array}\right| =2*\left| \begin{array}{ccc}13&1&2\\8&1&-2\\(11)/(2)& 1&(1)/(2)\end{array}\right| =2*\left| \begin{array}{ccc}13&1&2\\-5&0&-4\\(-5)/(2)& 0&(5)/(2)\end{array}\right| \\\\=2*(-1)*((-25)/(2)-(20)/(2)) =45\\`

`\Delta_2=\left| \begin{array}{ccc}2&13&2\\1&8&-2\\1&11&1\end{array}\right| \\\\\\=\left| \begin{array}{ccc}3&21&0\\3&30&0\\1&11&1\end{array}\right| \\\\\\=1*(90-63) =27\\`

`\Delta_3=\left| \begin{array}{ccc}2&1&13\\1&1&8\\1&2&11\end{array}\right| \\\\\\=\left| \begin{array}{ccc}0&-1&-3\\0&-1&-3\\1&2&11\end{array}\right| \\\\\\=0\\`

`\left\{\begin{array}{ccc}x=(\Delta_1)/(\Delta)=(45)/(9)=5\\\\y=(\Delta_2)/(\Delta)=(27)/(9)=3\\\\z=(\Delta_3)/(\Delta)=(0)/(9)=0\\\end{array}\right.`