1. May 13, 2010

### spoc21

1. The problem statement, all variables and given/known data

Solve the following systems and interpret the result geometrically
x - y - 2z - 3 = 0
2x - 3y - 3z + 15 = 0
x - 2y - z + 10 = 0

2. Relevant equations

3. The attempt at a solution

x - y - 2z - 3 = 0…………….(1)
2x - 3y - 3z + 15 = 0……….(2)
x - 2y - z + 10 = 0…………..(3)

first multiply equation (1) by -2
getting:

2x-2y-4z-6=0

Use elimination:

2x-2y-4z-6=0
-(2x - 3y - 3z + 15 = 0)
y-z-21=0

y-z=21

Elimination:

x-y-2z-3=0
-(x - 2y - z + 10 = 0)

y-z-13=0

==> y-z=13

Use elimination:

y-z=+21
y-z=13

Use elimination

(y-z=21)
-(y-z=13)
0=8

The answer is 0=number..This means that the system is inconsistent, and the planes never intersect.

I would really appreciate it if someone could take a look over my working, and point out any mistakes.

Thanks!

2. May 13, 2010

### Tedjn

Your working looks right to me, so the system is inconsistent like you said. However, be careful. In this case every pair of planes does intersect in a line. You can see this because parallel planes have proportionate coefficients for each independent variable and different constant terms. What is the actual (more precise) geometric interpretation?

3. May 13, 2010

### spoc21

Thanks Tedjn

So, there is an intersecting line? because I'm really confused; isn't 0=8 a false statement, meaning that the planes are neither parallel, nor they intersect. Is this an example of planes intersecting in pairs? could you please elaborate a little.

Thanks!

4. May 13, 2010

### Tedjn

Yes, the planes do intersect in pairs. Most books have a picture of this occurring but where the three planes do not intersect together at any point or line, so that the system has no (simultaneous) solution.

5. May 13, 2010

### spoc21

Thank you.

Just one more question: when it says to interpret the result geometrically, do I have to graph it? or is it just stating the facts that we discussed above?

6. May 13, 2010

### Tedjn

I believe just explaining the facts would be enough. If you are artistic, you might draw a simple picture illustrating how such pairwise intersections might look, but nothing accurate is probably required.