# How do you solve these equations? x+y+2z=5 x+2y+z=8 2x+3y-z=1

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### 2 Answers

We have to solve the system of linear equations:

x+y+2z=5 ...(1)

x+2y+z=8 ...(2)

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

(1) - (2)

=> -y + z = -3 ...(4)

(3) - 2*(1)

=> y - 5z = -9 ...(5)

(4) + (5)

=> -4z = -12

=> z = 3

y = -9 + 15 = 6

x = 5 - y - 2z = 5 - 6 - 6 = -7

**The solution to the system of equations is x = -7, y = 6 and z = 3**

x+y+2z=5 let this be equation (1)

x+2y+z=8 let this be equation (2)

2x+3y-z=1 let this be equation (3)

Solve the first 2 equations simultaneously to get a 4th equation:

(1)-(2):

(x+y+2z)-(x+2y+z) = 5-8

z-y=-3 let this be equation (4)

Solve the next 2 equations simultaneously to get a 5th equation(eliminated variable must be the same - in this case x):

2x(2)-(3):

2(x+2y+z)-(2x+3y-z)=2(8) - 1

(2x+4y+2z)-(2x+3y-z)=16-1

y+3z=15 let this be equation (5)

Solve equations (4) and (5) simultaenously:

(4)+(5):

(z-y)+(y+3z)=15+(-3)

4z=12

z=3*

*substitute this result into equation (4) or (5)

substituting into equation (4)

z-y=-3

3-y=-3

y=6**

**substitute both results found into equation (1), (2) or (3)

x+y+2z=5

x+6+2(3)=5

x=-7

Therefore, x=-7,y=6,z=3

For the second part, you could use a graphing software of some sort to be able to see the graphs more clearly. An example of one would be http://www.coolmath.com/graphit/. Just key in the equations and see the graph.