Question

How do you solve using gaussian elimination or gauss-jordan elimination, `6x+2y+7z=20`, `-4x+2y+3z=15`, `7x-3y+z=25`?

Answer 1

`x = 325/102`

`y= -165495/2601`

`z = 470/51`

Explanation:

`6x+2y+7z = 20`
`-4x +2y +3z = 15`
`7x - 3y +z = 25`

Gaussian elimination is a process of solving simultaneous equations in more than two variables by repeatedly adding and/or subtracting the equations.
Observing that there is an element `+2y` in each of the first two equations, if we subtract the middle equation from the first one we get
`10x+4z = 5`

If we add `3*`the second equation to `2*`the third equation

`-12x+6y+9z=45`
`14x-6y+2z=50`

we get `2x+11z = 95`

Using a similar process on these two equations in `x` and `z`
`10x+4z = 5`
`10x+55z = 475`
`51z = 470`

`z = 470/51`

`x = (5-4z)/10 = (255 - 1880)/510 =1625/510 = 325/102`

Select any one of the original equations to solve for `y`

`2y = 20 -6*(325/102) - 7*(470/51)`
`2y = (20*102*51 - 6*325*51 - 7*102*470)/(102*51)`

`y = (104040 - 99450 -335580)/5202`
`y= - 330990/5202 = -165495/2601`