Question:
a) Solve the system:
$$\begin{cases}x-y+z=10 \\ 2x+3y-2z=-21 \\ \frac{1}{2}x+ \frac{2}{5}y+ \frac{1}{4}z=-\frac{1}{2} \end{cases}$$
b) Give a geometric interpretation of the solution(s).
I tried doing part a) but I instantly got confused as I don't know what the question is asking me. For example: How would I solve this? What would I need to do first? Is there more than one solution to this?
As I tried to solve part b), I realized that I have solve to part a) before getting to part b). As well, I got more confused on the question of part b) as well. If anyone can help me out, that would be great. I am a new user so if this question is not that good of a question, then I apologize. Thanks.
Here is how I tried to get the answer. I still haven't gotten the answer because I feel that the way I'm doing it is incorrect. Can anyone check it over, and please help me out?
Thanks again.
Guide:
You should study a more systematic way using Gaussian elimination.
I see that you are trying to use substitution, that's fine too, but you should not substitute everything simultaneously as that would not get rid of variables. Let's do it one by one.
$$x=10+y-z$$
Now substitute this into the second and third equation.
$$2(10+y-z)+3y-2z=-21$$
$$\frac12(10+y-z)+\frac25y+\frac14z=-\frac12$$
Now you have two equations and two unknown, try to solve them, either by another substitution or elimination.
Spoiler: