Step 1

Consider the system of equations,

2x-y+z=-3...(1)

x-3y=2...(2)

x+2y+z=-7...(3)

In order to solve the given system of equations, first solve equation (2) for x in terms of y.

So, solving (2) for x, it gives

x-3y=2

\(\displaystyle\Rightarrow{x}={3}{y}+{2}\)

Step 2

Now substitute this value of x in equation (1) and (3) to get two equations in two variables. So, it gives

2(3y+2)-y+z=-3

6y+4-y+z=-3

5y+z=-7...(4)

And substituting value of x in (3), it gives

(3y+2)+2y+z=-7

5y+z=-9...(5)

Now subtract (4) from equation (5) and simplify further to get

(5y+z)-(5y+z)=(-9)-(-7)

5y+z-5y-z=-9+7

0=-2 False statement

Since the result on solving the equation (4) and (5) is not true, so there are no solutions of the given system of equations.

Step 3

Thus, the system of equations have no solutions and the system is inconsistent.

Consider the system of equations,

2x-y+z=-3...(1)

x-3y=2...(2)

x+2y+z=-7...(3)

In order to solve the given system of equations, first solve equation (2) for x in terms of y.

So, solving (2) for x, it gives

x-3y=2

\(\displaystyle\Rightarrow{x}={3}{y}+{2}\)

Step 2

Now substitute this value of x in equation (1) and (3) to get two equations in two variables. So, it gives

2(3y+2)-y+z=-3

6y+4-y+z=-3

5y+z=-7...(4)

And substituting value of x in (3), it gives

(3y+2)+2y+z=-7

5y+z=-9...(5)

Now subtract (4) from equation (5) and simplify further to get

(5y+z)-(5y+z)=(-9)-(-7)

5y+z-5y-z=-9+7

0=-2 False statement

Since the result on solving the equation (4) and (5) is not true, so there are no solutions of the given system of equations.

Step 3

Thus, the system of equations have no solutions and the system is inconsistent.