Question

Examine whether the following system of equations is consistent or inconsistent. If consistent, find the complete solutionx + y + 4z = 63x + 2y - 2z = 95x + y + 2z = 13

Answer 1

Answer:

Given system is consistent and have unique solution.

$x=\frac{13}{409} ,y=\frac{2860}{409},z=\frac{611}{409}$

Step-by-step explanation:

To Examine whether the following system of equations is consistent or inconsistent,first find out the rank of Augment matrix and coefficient matrix,if both rank are equal and equal to the number of variable than only the following system of equations are consistent.

Augment matrix

$\left[\begin{array}{cccc}1&1&4&13\\63&2&-2&13\\95&1&2&13\end{array}\right] \\\\\\R_{2} -> R_{2 }-63R_{1}\\\\R_{3} -> R_{3 }-95R_{1}\\\\\\\left[\begin{array}{cccc}1&1&4&13\\0&-61&-254&-806\\0&-94&-378&-1222\end{array}\right]\\\\\\R_{3} -> R_{3 }-\frac{94}{61} R_{2}\\\\\\\left[\begin{array}{cccc}1&1&4&13\\0&-61&-254&-806\\\\0&0&\frac{818}{61}&\frac{1222}{61} \end{array}\right]$

So

$Rank\left[\begin{array}{cccc}1&1&4&13\\63&2&-2&13\\95&1&2&13\end{array}\right]=\left[\begin{array}{cccc}1&1&4&13\\0&-61&-254&-806\\\\0&0&\frac{818}{61}&\frac{1222}{61} \end{array}\right]=3$

it is clear that rank of coefficient matrix is also 3.

So,given system is consistent and have unique solution.

Solution by Cramer's Rule:

$x=\frac{\triangle}{\triangle_{1}}\\\\\\y=\frac{\triangle}{\triangle_{2}}\\\\\\z=\frac{\triangle}{\triangle_{3}}$

$\triangle=\left|\begin{array}{ccc}1&1&4\\63&2&-2\\95&1&2\end{array}\right|=-818$

$\triangle_{1}=\left|\begin{array}{ccc}13&1&4\\13&2&-2\\13&1&2\end{array}\right|=-26$

$\triangle_{2}=\left|\begin{array}{ccc}1&13&4\\63&13&-2\\95&13&2\end{array}\right|=-5720$

$\triangle_{3}=\left|\begin{array}{ccc}1&1&13\\63&2&13\\95&1&13\end{array}\right|=-1222$

$x=\frac{-26}{-818} ,y=\frac{-5720}{-818},z=\frac{-1222}{-818}$

$x=\frac{13}{409} ,y=\frac{2860}{409},z=\frac{611}{409}$