Help with a system of equations with three variables?

Question

2x - 4y + 2z = -8
2x + y - z =10
3x - y + 2z=4

Solve

Answer 1

2x - 4y + 2z = -8
2x + y - z =10
3x - y + 2z=4
First step: Just use two equations and delete one variable. I'll choose to delete y.
2x + y - z =10
(+) 3x - y + 2z=4
_______________
5x +z = 14
Second step: Now you need to combine two OTHER equations to also delete a y.
2x - 4y + 2z = -8
2x + y - z =10
Since these can't cancel, you have to multiply the second by 4.
4(2x + y - z =10) =
8x+4y-4z=40
Then, use THIS equation with the other one.
2x - 4y + 2z = -8
(+) 8x+4y-4z=40
________________
10x -2z = 32
Now combine these two equations.
5x +z = 14
10x -2z = 32.
Again, multiply the first one by 2, so that it can cancel out.
2(5x +z = 14) =
10x + 2z = 28
And add these two equations.
10x -2z = 32.
10x + 2z = 28
_____________
20x = 60
x = 3.
Then plug this number back into one of your other equations... Like this one.
5x +z = 14
5(3) + z = 14
15 + z = 14
z = -1
Then plug the x value and the z value back into one of the ORIGINAL equations.
2x + y - z =10
2 (3) + y - (-1) = 10
6 + y +1 = 10
y +7 = 10
y = 3.
So your final answer is
x= 3
z = -1
y = 3.
Plug the numbers back in, and you'll see that they work (:

Answer 2

To begin, you are going to need to eliminate a variable. I would start with eliminating the "z" from the top and bottom equation by using the middle equation. First, take the 2nd equation and multiply everything times "2" that way you can have the z's match up nicely in all the problems. The second equation would then be ..

4x + 2y - 2z = 20

Now add the first and newly made 2nd equations.

(2x - 4y + 2z) + (4x + 2y - 2z) = -8 + 20
6x - 2y = 12

Now do the same for the new 2nd and the 3rd equations

(4x + 2y - 2z) + (3x - y + 2z) = 20 + 4
7x + y = 24

Now you have solve the two equations we just made for x. You will need to multiply the 2 * (7x + y = 24) so the y's will match up. You get 14x + 2y = 48. So ...

(6x - 2y) + (14x + 2y) = 12 + 48
20x = 60 so ... x = 3!! One part down!!

Now go back and plug in x to one of those equations to find y..

6(3) - 2y = 12 ... 18 - 2y = 12 ... -2y = -6 ... y = 3!! Two parts!!

Finally ... we have 2 of the three variables solved so we go back to one of the original equations -- any will do because you should get the same z for them all and fill in with our newly discovered values for x and y

2(3) + (3) - z = 10 .. 6 + 3 - z = 10 .. 9 - z = 10 .. z = -1

We have all three values!! x = 3, y = 3, and z = -1!

Check your answer by plugging in all three into the other equations to see if it works .. for example ..

2(3) - 4(3) + 2(-1) = -8 .. 6 - 12 - 2 = - 8 .. -8 = -8 TRUE!!!

Answer 3

Find x:
2x - 4y + 2z = - 8
2x = 4y - 2z - 8
x = 2y - z - 4

Find y plug x with 2y - z - 4:
2x + y - z = 10
2(2y - z - 4) + y - z = 10
4y - 2z - 8 + y - z = 10
5y = 3z + 18
y = 3/5z + 18/5

Find x plug y with 3/5z + 18/5
x = 2y - z - 4
x = 2(3/5z + 18/5) - z - 4
x = 6/5z + 36/5 - z - 4
x = 1/5z + 16/5

Find z plug x with 1/5z + 16/5 and y with 3/5z + 18/5:
3x - y + 2z = 4
3(1/5z + 16/5) - (3/5z + 18/5) + 2z = 4
3/5z + 48/5 - 3/5z - 18/5 + 2z = 4
2z = 4 - 6
2z = - 2
z = - 1

Find x plug z with - 1:
x = 1/5z + 16/5
x = 1/5(- 1) + 16/5
x = - 1/5 + 16/5
x = 3

Find y plug z with - 1:
y = 3/5z + 18/5
y = 3/5(- 1) + 18/5
y = - 3/5 + 18/5
y = 3

Answer: x = 3, y = 3, z = - 1

Proof:
2(3) - 4(3) + 2(- 1) = - 8
6 - 12 - 2 = - 8
- 8 = - 8

2(3) + 3 - (- 1) = 10
6 + 3 + 1 = 10
10 = 10

3(3) - 3 + 2(- 1) = 4
9 - 3 - 2 = 4
4 = 4

Answer 4

2x - 4y + 2z = - 8
4x + 2y - 2z = 20-----ADD
6x - 2y = 12

4x + 2y - 2z = 20
3x - y + 2z = 4----ADD
7x + y = 24

14x + 2y = 48
6x - 2y = 12----ADD
20x = 60
x = 3

42 + 2y = 48
2y = 6
y = 3

6 - 12 + 2z = - 8
2z = - 2
z = - 1

x = 3 , y = 3 , z = - 1

Answer 5

2x - 4y + 2z = -8......(1)
2x + y - z =10......(2)
3x - y + 2z=4......(3)
(1)+2(2): 6x-2y = 12......(4)
(3)+2(2): 7x+y = 24......(5)
(4)+2(5): 20x = 60
x = 30
y = -186
z = -136

Answer 6

x= -9, y=87, z=59...... Now you know both ends...you only need to figure out the middle, now that you know the answer.

Answer 7

Are you serious?