Question

use Cramer's method to solve the equation 2x-y+3z=9, x+y+z=6, x-y+z=2

Answer 1

Answer:

x=1, y=2, z=3

Step-by-step explanation:

The determinant made from the coefficients is:

$\displaystyle\Delta=\left|\begin{array}{ccc}2&-1&3\\1&1&1\\1&-1&1\end{array}\right|=2-1-3-3+2+1=-2$

The value of x is obtained by using this determinant with the x coefficients in the first column replaced by the values 9, 6, 2, and then dividing the result by the value of the coefficient determinant Δ above.  So...

$\displaystyle\left|\begin{array}{ccc}9&-1&3\\6&1&1\\2&-1&1\end{array}\right|=9-2-18-6+9+6=-2\\\\\Rightarrow x = \frac{-2}{\Delta}=\frac{-2}{-2} = 1$

Similarly for y, we replace the second column of y coefficients with the values 9, 6, 2 to get:

$\displaystyle\left|\begin{array}{ccc}2&9&3\\1&6&1\\1&2&1\end{array}\right|=12+9+6-18-4-9=-4\\\\\Rightarrow y=\frac{-4}{\Delta}=\frac{-4}{-2}=2$

Finally, for z we have:

$\displaystyle\left|\begin{array}{ccc}2&-1&9\\1&1&6\\1&-1&2\end{array}\right|=4-6-9-9+12+2=-6\\\\\Rightarrow z=\frac{-6}{\Delta}=\frac{-6}{-2}=3$