Systems of Equations in Three Variables?

Question

I know this might be a lot of work if you're doing it on paper, but if you can just give me the answers to:

2x-y+2x=15
-x+y+z=3
3x-y+2z=18

a+b=2
-b+c=3
a+2c=10

2x+3y+4z=2
5x-2y+3z=0
x-5y-2z=-4

2x-5y+z=5
3x+2y-z=17
4x-3y+2z=17

x+y+z=6
z=x+y
x=z+1

Answer 1

2x - y + 2z = 15- - - - - -Equation 1
-x + y + z = 3- - - - - - - Equation 2
3x - y + 2z = 18- - -- - - Equation 3
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Elimination of y Equation 1 and equation 2

2x - y + 2z = 15
-x + y + z = 3
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x + 3z = 18 . . Equation 4

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Multiply equation 3 by - 1

3x - y + 2z = 18
- 1(3x) - 1(-y) + (- 1)(2z) = - 1(18)
-3x + y - 2z = - 18. . .New equation 3

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Elimination of y and z equation 1 and new equation 3

2x - y + 2z = 15
-3x + y - 2z = - 18
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- x = - 3

-1x = -3/-1

x = 3

The answer is x = 3

Insert the x value into equation 4

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Solving for z

x + 3z = 18

3 + 3z = 18

3 + 3x - 3 = 18 - 3

3z = 15

3z/3 = 15/3

z = 5

The answer is z = 5

Insert the z value into equation 1

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Solving for y Equation 1

2x - y + 2z = 15

2(3) -y + 2(5) = 15

6 - y + 10 = 15

16 - y = 15

16 - y - 16 = 15 - 16

- y = - 1

- 1y = - 1

-1y/-1 = - 1/- 1

y = 1

The answer is y = 1

Insert the y value into equation 1

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Check for equation 1

2x - y + 2z = 15

2(3) - 1 + 2(5) = 15

6 - 1 + 10 = 15

5 + 10 = 15

15 = 15

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Check for equation 2

- x + y + z = 3

- 3 + 1 + 5 = 3

-2 + 5 = 3

3 = 3

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Check for equation 3

3x - y + 2z = 18

3(3) - 1 + 2(5) = 18

9 - 1 + 10

8 + 10 = 18

18 = 18

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The solution set is { 3, 1, 5 }

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Answer 2

I was going to explain how to do them, but you say you just want the answers and I'm sure someone else will give them while I'm typing out a painstaking explanation, so I won't. Believe me, it's more satisfying to find out how to do a couple and then use the same method to do the rest yourself.

Answer 3

1)x=3
y=-3
z=3
2)c=5
b=2
a=0
3)x=2
y=2
z=-2
4)x=5
y=1
z=0
5)x=4
y=-1
z=3