Question

Find the linear equation?

I've been stuck on this problem for hours. I feel like I've tried every possible option and keep reaching a dead end! I don't need the answer as much as I need to know how you solved the problem!

10x + 2y - 14z = 8
-x - 2y - 4z = -1
-12x - 6y + 6z = -12

Answer 1

This set of equations has no solution.

Explanation:

`E1: 10x+2y-14z=8`
`E2: -x-2y-4z=-1`
`E3:-12x-6y+6z=-12`

The first thing I'll do is divide E3 by 6 to make it a bit easier to work with:

`E3:-2x-y+z=-2`

Let's now solve `E3` for `z` and substitute into `E1 and E2`:

`E3: z=2x+y-2`

`E1: 10x+2y-14(2x+y-2)=8`

`=> -18x-12y=36`

`E2: -x-2y-4(2x+y-2)=-1`

`=> -9x-6y=-9`
`=>-18x-12y=-18`

And here's where things get unsolvable, because what we have in these two equations are no points that are similar - the lines themselves are parallel.

There is no set of `(x,y,z)` points that will satisfy all three equations.

Answer 2

In support of Parzival S. solution

There is a contradiction in the system. So either the question is mistyped or there is no solution.

Explanation:

I used something called Gaussian Elimination. Really it is the same process used for simultaneous equation but it just looks different.

You will need to zoom in to see clearly what I have done.