Good evening, everyone!
There are two problems that I'm having difficulty figuring out on my homework. I'd really appreciate the help!
(1) 2x - y = 2
-x + 3y + 2z = 1
x + 2y + 2z = -1
I created an augmented matrix from this and row reduced to the point where I have:
|1 -3 -2 | -1|
|0 1 4/5 | 0|
|0 5 4 | 0|
Not sure if I did incorrect calculations or what, but I have no idea where to go from here (provided I did the correct calculations up to this point). Please help me out!
(2) w + 2x - y - 2z = -1
x - y = -1
-w + 2x + z = -1
Here I also row reduced but only went through one round of reducing before coming to the conclusion that there's no solution...Yes or no?
Please, please help -- I really am quite lost with both of these!
-Lux
Hello, Lux!
Multiply the second equation by -1 and list it first: .
We have: .
The bottom row gives us: . . . . The system is inconsistent (no solution).
We have: .
. . .
The system has an infinite number of solutions.
We have: . .
On the right, replace with a parameter
. . . .
This represents all the solutions to the system,
. . one for each value of
Edit: Too fast for me, Danny . . .
. . . . And I like your solutions!
.
Thanks guys! I really am having a lot of troubles with Gauss-Jordan.
I'm so sorry, but I accidently typed the last problem incorrectly (2) --- the first line of that equation should read 2x-y = -2. I did the problem again with the correct equation and row reduced to the point where I have:
1 -3 -2 | -1
0 1 4/5 | 0
0 5 4 | 0
I know that I need to change the -3 and the 5 in the second column to zero, but I don't know what to do from here. And I have a feeling I may have miscalculated again.
Next, I'm dealing with a bit of a monsterous problem (it is a word problem, and we need to create a linear equation and subsequently a matrix from the numbers given). I got help from a math tutor at school setting up the equations involved, but again, I know that I am miscalculating something while I'm row reducing because I'm not getting the right answer. Gauss-Jordan is particularly frustrating, because if you mess up even the tiniest little thing, everything is affected. Here are my two equations:
100,000x + 70,000y +40,000z = 1,600,000
100x +50y = 1000
20,000x + 5,000y = 150,000
100,000x + 70,000y +40,000z = 1,140,000
100x +50y = 700
20,000x + 5,000y = 110,000
I have pages of scratch-work for this, nothing makes sense anymore.
Thirdly, when a problem wants to know if the answers are inconsistent or not, what does this mean? How do you know if something is inconsistent? Do I have to keep row-reducing until I can't reduce anymore...and that means it's inconsistent? Please help, I'm way lost.