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# Solve the system. If the system does not have one unique solution, also state whether the system is onconsistent or whether the equations are dependent. 2x-y+z=-3 x-3y=2 x+2y+z=-7

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asked 2021-03-29
Solve the system. If the system does not have one unique solution, also state whether the system is onconsistent or whether the equations are dependent.
2x-y+z=-3
x-3y=2
x+2y+z=-7

## Answers (1)

2021-03-31
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.

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