Question

How do you solve using gaussian elimination or gauss-jordan elimination, `x+y+z=1`, `3x+y-3z=5` and `x-2y-5z=10`?

Answer 1

`((x),(y),(z)) = ((6),(-7),(2))`

Explanation:

The linear equation:
`x+y+z =1`
`3x+y-3z=5`
`x-2y-5z=10`
can be written in Matrix form as:
`(x, y, z)((1,1,1),(3,1,-3),(1,-2,-5)) = ((1), (5), (10))`
Let's now use gauss elimination:
`((1,1,1, |, 1),(3,1,-3, |, 5),(1,-2,-5, |, 10))`

`color(red)(R_2=-3R_1+R_2)` `((1,1,1,|, 1),(0,-2,-6,|, 2),(1,-2,-5, |, 10))`

`color(red)(R_3 =-R_1+R_3)` `((1,1,1,|, 1),(0,-2,-6,|, 2),(0,-3,-6, |, 9))`

`color(red)(R_3 =-3/2R_2+R_3)` `((1,1,1,|, 1),(0,-2,-6,|, 2),(0,0,3, |, 6))`
Now what we have is:
`x+y+z= 1` --------------------------------------------------------- (1)
`" " -2y-6z = 2` ---------------------------------------------------------(2)
`" " 3z = 6`; `z= 2` substitute this into (2) above solve for y
`y = -7` sub for y and z into (1) `x= 6`