Chapter 11.2, Problem 51AYU

To determine

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

Answer to Problem 51AYU

Solution:

No solutions, Inconsistent system.

Explanation of Solution

Given:

$2x-2y-2z=2$

$2x+3y+z=2$

$3x+2y=0$

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:

$2x-2y-2z=2$

$2x+3y+z=2$

$3x+2y=0$

Subtracting first equation from the second equation we get,

$5y+3z=0$

Now we get,

$3x+2y=0$

$5y+3z=0$

Or

$5\left(3x+2y\right)=0$

$2\left(5y+3z\right)=0$

Or

$15x+10y=0=10y+6z$

$15x=6z$

$5x=2z$

Then from the first equation we get,

$2x-2y-5x=2$

$-3x-2y=2$

$3x+2y=-2$

Thus we get,

$3x+2y=0$

$3x+2y=-2$

Both the above equations have the same LHS but different RHS (0 and $-2$).

This is not possible.

Thus the given system of equations has no solutions and hence is inconsistent.