Finding the solution of three planes

[gracelette]

Homework Statement

given the planes with equations:
x + y + 7z = -7
2x + 3y + 17z = -16
x + 2y + (a^2 + 1) z = 3a

find values for the constant a for which:
-there are no solutions
-the planes meet in a line. in this case find the parametric equation of the line
-meet at a point. then find the parametric equation of the point.

Homework Equations

gaussian elimination? I don't think there are any. =S

The Attempt at a Solution

I put the planes into an augmented matrix, and then into echelon form, and got (I think its correct) the general solution, which is attached. From the I can read off the equation of the line, right? The parts I'm really stuck on are parts one and three. I realize that for the planes to meet in a point, there has to be 3 pivots but beyond that I'm a bit stumped.

Well actually, I have a bit of a clue for part one, that a =/= 3 because then 0z = 18 which is inconsistent. is that right? But for part 3, I'm not sure how to do it, really.

 

[Mark44]

Homework Statement

given the planes with equations:
x + y + 7z = -7
2x + 3y + 17z = -16
x + 2y + (a^2 + 1) z = 3a

find values for the constant a for which:
-there are no solutions
-the planes meet in a line. in this case find the parametric equation of the line
-meet at a point. then find the parametric equation of the point.

Homework Equations

gaussian elimination? I don't think there are any. =S

The Attempt at a Solution

I put the planes into an augmented matrix, and then into echelon form, and got (I think its correct) the general solution, which is attached. From the I can read off the equation of the line, right? The parts I'm really stuck on are parts one and three. I realize that for the planes to meet in a point, there has to be 3 pivots but beyond that I'm a bit stumped.

Well actually, I have a bit of a clue for part one, that a =/= 3 because then 0z = 18 which is inconsistent. is that right? But for part 3, I'm not sure how to do it, really.

You have a mistake. In your first augmented matrix, the bottom row is 0 0 a² - 9 | 3a + 9

Right after that, you say let a = -3i. There are two real values that make a² - 9 equal to 0.

For your other questions, there will be no solutions if the last equation is 0z = k, with k not equal to 0.
There will be multiple solutions (points along a line) if the last equation is 0z = 0.
Finally, there will be a unique solution (a single point) if the last equation is kz = <whatever>, with k not equal to 0.

 

[gracelette]

Ooops. I guess this is what happens when I try do maths at obscene times of day. Thankyou, i worked it out now!