Dayv O. answered • 09/29/21
Caring Super Enthusiastic Knowledgeable Algebra Tutor
if determinant of matrix is not zero, then three planes intersect at one point
if determinant is zero, but in eliminating variables find 0=0 on one equation as last step
then planes meet in a line
if determinant is zero and in eliminating variables find 0=k on one equation as last step
then planes meet in three parallel lines
the above assumes no two nor all three planes are parallel, planes are parallel if the coefficients
of the variables of one plane are in ratio to those of another plane. If two planes are parallel then the third plane will intersect both if it is not a parallel plane.
Correction (I copied 11 as -11), now the following is correct.
the matrix is zero so there is no single point that is the solution.
according to the rules I give, the first system results in three parallel lines since I find 0=-1 when eliminating z after eliminating x and y, and in the second system 0=0 in eliminating z after eliminating x and y. and therefore the second system results in one line of intersection.
