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I am completing a practice questions sheet for the topic "systems of linear of equations" and I'm having trouble on one of the questions.

1. Consider the system of equations $$\begin{aligned} x + 2y - z &= -3 \\\ \end{aligned}$$ $$\begin{aligned} 3x + 5y + kz &= -4 \\\ \end{aligned}$$ $$\begin{aligned} 9x + (k+13)y + 6z &= 9 \\\ \end{aligned}$$ a) Express these equations as an augmented matrix

which I think is: $$ \left[\begin{array}{rrr|r} 1 & 2 & -1 & -3 \\ 3 & 5 & k & -4 \\ 9 & (k+13) & 6 & 9 \end{array}\right] $$

b) Show that this matrix can be row-reduced to

$$ \left[\begin{array}{rrr|r} 1 & 2 & -1 & -3 \\ 0 & 1 & -k-3 & -5 \\ 0 & 0 & k^2-2k & 5k+11 \end{array}\right] $$

c) hence answer the following questions

for what value(s) of k does the system have

. no solutions: I think when k=0, k=2 and k does not = -11/5

. a unique solution: when k= any real number other than 2,0 and -5/11

. infinitely many solutions: I couldn't find anything for this so I believe that this system doesn't have infinite solutions (correct me if I'm wrong)

the question that I'm particularly stuck on is:

"Each of these equations represents a plane. In each case in (the above question), give a geometric description of the intersection of the three planes."

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You are correct in everything you said — indeed the matrix cannot have infinitely many solutions. Sometimes questions are like that! They ask for “how many” and the answer ends up being $0$.

Onto the second part, if you imagine two lines in the plane, they are either parallel or intersect. With planes in space, it is the same thing:

  • If any two planes are parallel, then there is no triple point of intersection.

  • It is possible that all of the planes pairwise intersect, but there is no triple intersection point (think of three faces in a triangular prism).

  • More often than not, the planes don’t have any parallel stuff happening and they all pass through the same point

  • Sometimes all three planes pass through a line, in which case you have infinitely many solutions

  • Final case is when they are all just the same plane, so you get infinitely many solutions again

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