Question

Solve the following system of equations for all three variables. - x+y+4z=6 x+5y+2z=0 x+8y-8z=-6

Answer 1

We have the following system of equations:

`x+y+4z=6`
`x+5y+2z=0`
`x+8y-8z=-6`

If we solve for x for the first equation we got:

`x=6-y-4z\text{ (1)}`

Now we can replace in the second equation and we got:

`6-y-4z+5y+2z=0`
`4y-2z=-6`

And solving for y we got:

`y=(2z-6)/(4)\text{ (2)}`

Replacing the equations (1) and (2) into the final equation we got:

`6-((2z-6)/(4))-4z+8((2z-6)/(4))-8z=-6`
`6-(z)/(2)+(3)/(2)-4z+4z-12-8z=-6`
`(17)/(2)z=6+6-12+(3)/(2)`
`z=(3)/(17)`

Now we can solve for y and x like this:

`y=(2(3/17)-6)/(4)=-(24)/(17)`
`x=6-(-(24)/(17))-4((3)/(17))=(114)/(17)`

And the final solution would be:

x= 114/17, y= -24/17, z= 3/17