Question

Solve the system. State whether it is inconsistent or has infinitely many solutions. If the system has infinitely many solutions, write the solution set with z arbitrary 4x + 4y - 8z = 5 3x + y - Z = 6 -X-y + 225 Select the correct choice below and will in any answer boxes within your choice. at O A. There is one solution. The solution set is ( CD) (Simplify your answers ) O B. There are infinitely many solutions. The solution set is {OL-)), where z is any real number, (Simplify your answers. Type expressions using z as the variable) C. The system is inconsistent. The solution set is dy ade

Answer 1

Solution-

Consider the system of equations

4x + 4y - 8z = 5

3x + y - z = 6

- x - y - 2z = 5

The augumented matrix corrosponding to this system of equations is

[A|B] =

$\left \[ \left.\begin{matrix} 4& 4&-8 \\ 3& 1 &-1 \\ -1&-1 & 2 \end{matrix}\right| \right.\left.\begin{matrix} 5\\ 6\\ 5 \end{matrix}\right\]$

$R_3\rightarrow R_3+R_1/3\, and\, R_2\rightarrow R_2-3R_1/4$

$\Rightarrow \left \[ \left.\begin{matrix} 4& 4&-8 \\ 3-3& 1-3 &-1+6 \\ -1+1&-1 +1 & 2-2 \end{matrix}\right| \right.\left.\begin{matrix} 5\\ 6-15/4\\ 5+5/4 \end{matrix}\right\]$

$\Rightarrow \left \[ \left.\begin{matrix} 4& 4&-8 \\ 0& -2 &5\\ 0&0& 0 \end{matrix}\right| \right.\left.\begin{matrix} 5\\ 9/4\\ 25/4 \end{matrix}\right\]$

This the row reckon form of the matrix.

Here we can observe that rank of [A] is 2 while rank of [A|B] = 3.

Since [A] < [A|B] .

So there is no solution of the system of equations.

Hence, The system is inconsistent. The solution set is empty (option - C).