Question

does the following system of equations have a solution if so find it. 2x+y+z= 4 x-y+3z= -2 -x+y+z= -2

Answer 1

I'd suggest using "elimination by addition and subtraction" here, altho' there are other approaches (such as matrices, substitution, etc.).

Note that if you add the 3rd equation to the second, the x terms cancel out, and you are left with the system

 - y + 3z = -2
   y  + z  =  -2
-----------------
         4z = -4, so z = -1.

Next, multiply the 3rd equation by 2:  You'll get -2x + 2y + 2z = -2.

Add this result to the first equation.  The 2x terms will cancel, leaving you with the system

2y + 2z = -2
  y  +  z  = 4

This would be a good time to subst. -1 for z.  We then get:

-2y - 2 = -2.  Then y must be 0.  y = 0.

Now subst. -1 for z and 0 for y in any of the original equations.

For example, x - (-1) + 3(0) = -2, so x + 1 = -2, or x = -3.

Then a tentative solution is     (-3, -1, 0).

It's very important that you ensure that this satisfies all 3 of the originale quations.