How would I solve these two equations?

Question

thephysicsman · Answer

$$E_{1} = x-y-z = 3 $$

$$E_{2} = x-10y+10z = 0 $$

thephysicsman · Answer

@Redcan if you have time

thephysicsman · Answer

this is what I did to start

anonymous · Answer

you can use lin. alg. $$\left(\begin{matrix}1&-1&-1\1&-10&10\0&0&0\end{matrix}\mid\begin{matrix}3\0\0\end{matrix}\right)$$

anonymous · Answer

again, you'll have at least one free param.

thephysicsman · Answer

$$E_{1}-E_{2} = (x-y-z)-(x+10y-10z) = 9y-11z = 3 $$

This is what I tried to attempt to do. eliminate one of the variables but it's taking forever... 

$$E_{1} = x-y-z= 3$$
$$E_{2} = 9y-11z = 3 $$
$$E_{3} = x-10y+10z =0$$

anonymous · Answer

that will work, but if $$E_3$$ is a linear comb. of the other equations, you will eliminate it in the end.

anonymous · Answer

$$E_3 = aE_1 + bE_2$$ so linear dependent.

thephysicsman · Answer

getting back to what you said in the other question, guess i'll have to chose a parameter...

anonymous · Answer

yep!

anonymous · Answer

and the way we have the matrix set up, letting z be the param might be easiest. $$z=s,\quad s\in\mathbb{R}$$

thephysicsman · Answer

like when I first saw this, I didn't understand the whole method of constantly adding equations.. and trying to remove variables when there's three of them..

anonymous · Answer

If you have more unknowns then equations you will have a parameter.  The system isn't fully determined.  Need three ind. eqn. for 3 unknowns.

anonymous · Answer

I just add the trivial eqn to make the matrix look nice. it's not required.  If a column doesn't have a pivot, assign it a parameter.  (pivot ~ to a leading one)

thephysicsman · Answer

Sorry, so another thing  about the first question I posted. 

I know these two are linear dependent. 

$$E_{1} = 8x-5y = 20 $$

$$E_{2} = -16x+10y = -40$$ 

y = t 

So i noticed that 

$$E_{1} = -\frac{ 1 }{ 2 }E_{2}$$ 

how would I go about writing the solution to this?

anonymous · Answer

if you set y=t, then $$8x-5y=20 \iff x=20/8+5t/8$$

anonymous · Answer

So any point $$(20/8+5t/8, t), \quad t\in\mathbb{R}$$

thephysicsman · Answer

Interesting:

thephysicsman · Answer

Thank a-lot for you help!

anonymous · Answer

np. Notice in our 2d problem with one param. the solution set forms a line.

anonymous · Answer

two free parameters ---> a plane. etc.