Chapter 11.2, Problem 55AYU

To determine

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

Answer to Problem 55AYU

Solution:

Inconsistent system of equations, no solutions.

Explanation of Solution

Given:

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

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

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

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:

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

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

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

From above we get,

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

$3y=4x+2z$

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

So we get,

$6y=6x+9z-18$

$6y=8x+4z$

$6y=4x+14z+2$

Now,

$6x+9z-18=8x+4z$

$2x-5z=-18$

Again

$8x+4z=4x+14z+2$

$4x-10z=2$

Or

$2x-5z=1$

We get two equations in $x$ and $z$ with the same LHS $\left(2x-5z\right)$ but different RHS $-18$ and 1.

This is not possible.

Thus above system of equations contains no solutions.

Therefore this is an inconsistent system.