Chapter 11.1, Problem 47AYU

To determine

To find: The solution of the given system of equations.

Answer to Problem 47AYU

Consistent, infinitely many solutions.

Explanation of Solution

Given:

$x-y-z=1$

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

$3x-2y-7z=0$

Calculation:

The system of equations is:

$x-y-z=1$

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

$3x-2y-7z=0$

Adding first and second equation we get,

$\left(x-y-z\right)+\left(-x+2y-3z\right)=1-4$

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

$y-4z=-3$

$y=4z-3$

Adding second and third equation we get,

$\left(-x+2y-3z\right)+\left(3x-2y-7z\right)=-4+0$

$-x+2y-3z+3x-2y-7z=-4$

$2x-10z=-4$

$x-5z=-2$

$x=5z-2$

Thus the solution set to the given system of equations is:

$\left\{\left(5z-2,4z-3,z\right):z\text{ is real}\right\}$.

Since $z$ can be any real number the number of solutions to this system of equations is infinite.

Hence the given system of equations is consistent and has infinite many solutions.