# Given the points A(1,1,2), B(1,2,1), C(2,1,1) which lie in plane x+ay+bz+c=0, what is the plane?

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The points A(1,1,2), B(1,2,1) and C(2,1,1) lie in plane x+ay+bz+c=0

To determine the value of a, b and c, solve the set of equations:

1 + a + 2b + c = 0 ...(1)

1 + 2a + b + c = 0 ...(2)

2 + a + b + c = 0 ...(3)

(1) - (3)

=> -1 + b = 0

=> b = 1

(1) - (2)

=> -a + b = 0

=> -a + 1 = 0

=> a = 1

Substitute a = 1 and b = 1 in (3)

=> 2 + 1 + 1 + c = 0

=> c = -4

**The equation of the plane is x + y + z - 4 = 0**

To find the plane, we can first set up a systems of equations and solve for a, b, c.

Put the corresponding numbers into the places of x, y, z in the equation:

For point A: `1+1a+2b +c=0`

For point B: `1+2a+1b+c=0`

For point C: `2+1a+1b+c=0`

Now we can isolate variables by subtracting equations.

(Note: Other methods for solving include matrices and substitution)

By subtracting the equations for points B and C, we can isolate and solve for variable a:

`1+2a+1b+c=0`

minus

`2+1a+1b+c=0`

--------- equals

`-1+a=0`

` ` `a = 1`

We can solve for variable b in the same way, subtracting the equations for points A and C:

`1+1a+2b +c=0`

`2+1a+1b+c=0`

---> `-1+b=0`

`b=1`

Then we can plug the values for a and b (in any equation), and solve for c:

`1+1(1) + 2(1) + c = 0`

`4+c=0`

`c=-4`

The plane is:

`x+y+z-4=0`