Posted by **Anonymous** on Tuesday, June 17, 2014 at 7:51pm.

Solve the system of linear equations

2x+y-3z=4

4x+2z=10

-2x+3y-13z=-8

*im so lost (all these numbers are in one problem)

- MATH -
**Reiny**, Tuesday, June 17, 2014 at 8:10pm
if you divide the 2nd by 2, all your equations start with 2x. That is good, since we could add or subtract them to eliminate the x

2x+y-3z=4

2x+z=10

-2x+3y-13z=-8

subtract 1st from the 2nd

-y + 4z = 6 , #4

add the 2nd and 3rd

3y - 12z = 2 , #5

#4 times 3---> -3y + 12z = 18

#5 as is -----> 3y - 12z = 2

add them: 0 = 20 , which is a contradiction

AHHH, there is no unique solution.

If you are studying equations of planes, you should categorize this as one of the special cases.

- MATH -
**Damon**, Tuesday, June 17, 2014 at 8:15pm
well, the easy way is to google Gauss Jordan reduction calculator.

http://www.gregthatcher.com/Mathematics/GaussJordan.aspx

otherwise do it by elimination

First eliminate z from first two:

2x+y-3z = 4 times 2 = 4x+ 2y-6z = 8

4x+0y+2z=10 times 3 = 12x+0y+6z = 30

add the two

16 x + 2 y = 38

or

8 x + y = 19

Now eliminate z from the last 2

multiply second by 13 and last by 2

52 x + 0y + 26 z = 130

-4 x + 6 y -26 z = -16

------------------------- add the two

48 x + 6 y = 114

or

24 x + 3 y = 57

Now you have the two equations in x and y

8 x + 1 y = 19

24 x + 3 y = 57

multiply first by 3

24 x + 3 y = 57

Oh , my, system can not be solved, determinant is zero, two lines the same.

- MATH -
**Reiny**, Tuesday, June 17, 2014 at 8:35pm
Your solution is illustrated by

http://www.google.com/search?q=intersection+of+3+planes&tbm=isch&imgil=VKeuvId73BKgcM%253A%253Bhttp%253A%252F%252Ft2.gstatic.com%252Fimages%253Fq%253Dtbn%253AANd9GcTP5OK2B8xVEFseHgCh5S25ao2K4pX-fClEpupY9ltfPRK6cZC8NA%253B332%253B265%253BL5ht1c6dmTcU5M%253Bhttp%25253A%25252F%25252Fstweb.peelschools.org%25252Fjfsweb%25252Fmga4u%25252Funit3%25252Flesson7.html&source=iu&usg=__Epsu58j8aAIOLuG__2toxIf0TSk%3D&sa=X&ei=SN6gU9wz2KPIBJaOgPAK&sqi=2&ved=0CCkQ9QEwAg&biw=1680&bih=936#imgdii=_

look at the 2nd picture, top left

Your planes intersect in pairs, each one giving us a parallel line as the intersection of the planes, two at a time.

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