Question

How do you solve the system of equations `4x+2y+z=7, 2x+2y+4z=-4, x+3y-2z=-8`?

Answer 1

`x=47/11`
`y=(-51)/11`
`z=(-9)/11`

Explanation:

Given -

`4x+2y+z=7`----------------(1)
`2x+2y+4z=-4`-------------(2)
`x+3y-2z=-8`----------------(3)

Take the first two equations and eliminate `y`

`4x+2y+z=7`----------------(1)
`2x+2y+4z=-4`-------------(2) -------- `(1)-(2)`
`2x-3z=11` -------------(4)

Take equations (1) and (3)
We have to elimiate `y` term

`4x+2y+z=7`----------------(1) -------`xx3`
`x+3y-2z=-8`----------------(3) -------`xx2`

`12x+6y+3z=21` ------------(5)
`2x+6y-4z=-16`-------------(6) ----- `(5)-(6)`
`10x+7z=37` ----------------(7)

Take equations (4) and (7)

`2x-3z=11` -------------(4) --------`xx5`
`10x+7z=37` ----------------(7)

`10x-15z=55` ---------(8)
`10x+7z=37`-----------(7) ------`(8)-(7)`
`-22z=18`
`z=-18/22`

Plugin >`z=-18/22` in equation (7)

`10x+7(-18/22)=37`
`10x-126/22=37`
`10x=37+126/22=(814+126)/22=940/22=470/11`
`x=470/11xx1/10=470/110=47/11`
`x=47/11`

Plugin >`z=-18/22` and >`x=47/11` in equation (1)

`4x+2y+z=7`----------------(1)
`4(47/11)+2y-18/22=7`
`4(47/11)+2y-9/11=7`
`188/11+2y-9/11=7`
`(188-9)/11+2y=7`
`179/11+2y=7`
`2y=7-179/11=(77-179)/11=(-102)/11`
`y=(-102)/11xx1/2=(-102)/22=`
`y=(-51)/11`

`x=47/11`
`y=(-51)/11`
`z=-18/22=(-9)/11`