Question

How do you solve using gaussian elimination or gauss-jordan elimination, `3w-x=2y + z -4`, `9x-y + z =10`, `4w+3y-z=7`, `12x + 17=2y-z+6`?

Answer 1

`w = -183/735`

`x = -111/49`

`y = 27297/1470`

`z = 71967/1470`

Explanation:

Gaussian elimination involves the gradual elimination of variables from the set of equations until values can be found for each variable.
`3w-x=2y+z-4`
`9x-y+z=10`
`4w+3y-z=7`
`12x+17=2y-z+6`

First reorder the terms in all equations so that they all have the same order of variables.
`3w-x-2y-z=-4`
`9x-y+z=10`
`4w+3y-z=7`
`12x-2y+z=-11`

Adding the first two equations together gives
`3w+8x-3y=6`
Adding the second pair of equations gives
`4w+12x+y =-4`
Adding the middle pair of equations gives
`4w+9x+2y =17`

In a similar process, add three times the middle equation to the first equation
`15w+44x = -6`
Take twice the middle equation from the third equation
`-4w-15x=9`

Now add four times the first equation to fifteen times the second equation
`176x - 225x = -24 +135`
`-49x = 111`

`x = -111/49`

Substitute this back into one of the final pair of equations to get `w`
`15w = -6 -(-111/49) = -183/49`
`w = -183/735`

`2y = 17 -4(-183/735) -9(-111/49) = (12495 -183+14985)/735`
`y = 27297/1470`

`z = 10-9x+y = 10-9(-111/49) +27297/1470`
`=(14700+29970+27297)/1470 = 71967/1470`