Planar Intersections (Answer Check)

[spoc21]

Homework Statement

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

Homework Equations

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!

 

[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?

 

[spoc21]

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?

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!

 

[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.

 

[spoc21]

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.

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?

 

[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.