Question

How do I solve this system by elimination?
(Steps would be very helpful)

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

Answer 1

Okay, so to solve by elimination you have to solve one system in terms of the other variables and then plug and go and keep doing that till you're done. So, to start, I'm going to pick the 3rd equation and solve it for x

x+2y+z=4 We subtract 2y and z to the other side and get
x=4-2y-z Now we have a value for x. and we can plug that in to a different equation and solve for a new variable. I'll choose the 2nd equation and z this time.

x+y+2z=4 Now we can plug in the values for x and get
4-2y-z+y+2z=4 Again we want to get all that stuff over to the right side except our variable of interest, this time... z.
Start by combining like terms and you get:
-y+z+4=4 We subtract the 4 and add the y and have
z=y Then we plug this AND the x information in to the first equation.

3x-y-z=3 Plugging in values for z and x:
3(4-2y-z)-y-y=3 Now we still have a z from our equation for x, so let's plug in our known value for z and get
3(4-2y-y)-y-y=3 Now we can distribute that 3 across the parenthetical case:
12-6y-3y-y-y=3 And combine like terms:
12-11y=3 subtract our 12
-11y=-9 divide by -11
y=9/11 or about .8181 if you prefer.

Now we know exactly what y is, so we go back up to our other solved equations and plug this number in. The second set of equations told us z=y so we know that z also equals 9/11 or .8181
Finally we plug in to our x=4-2y-z equation all of our y's and z's true value and it will tell us x.
x=4-2(9/11)-9/11 simplifying we get
x= 17/11 or about 1.5454

Now we just put our information together and say:
x= 17/11 or approx 1.5454
y= 9/11 or approx 0.8181
z= 9/11 or approx 0.8181