Chapter 11.2, Problem 56AYU

To determine

To find: The solution to the given system of equations using matrices.

Answer to Problem 56AYU

Solution:

Inconsistent system of equations, no solutions.

Explanation of Solution

Given:

$3x-2y+2z=6$

$7x-3y+2z=-1$

$2x-3y+4z=0$

Formula used:

To solve a system of three equations in $x,y$ and $z$ using matrices:

Step 1: Write the corresponding matrix associated with the system of equations.

Step 2: Use elementary row operations to get equivalent matrix of the form:

$\left[\begin{array}{ccc}1& a& b\\ 0& 1& c\\ 0& 0& 1\end{array}|\begin{array}{c}d\\ e\\ f\end{array}\right]$; where $a,b,c,d,e,f$ are constants.

Step 3: Solve for $x,y$ and $z$.

Calculation:

$3x-2y+2z=6$

$7x-3y+2z=-1$

$2x-3y+4z=0$

From above we get,

$3x+2z=2y+6$

$3y=7x+2z+1$

$3y=2x+4z$

So we get,

$7x+2z+1=2x+4z$

$7x-2x+2z-4z+1=0$

$5x-2z+1=0$

From above we get,

$6y=9x+6z-18=14x+4z+2=4x+8z$

$5x-2z=-20$

$10x-4z+2=0$

$5x-2z=-1$

We get two equations in $x$ and $z$ with the same LHS $5x-2z$ but different RHS $-20$ and $-1$.

This is not possible.

Thus above system of equations contains no solutions.

Therefore this is an inconsistent system.