I am stuck with this one problem: Find annual production (output) of coal and steel if total annual production is $20 million.
Know:
1st: I created a table to help me understand this problem more
2nd: I wrote 2 systems of equations and then rewrote them into homogeneous linear system of equations
c= 0.30c + 0.70s --> 0= -0.70c + 0.70s
s= 0.80c + 0.20s --> 0= 0.80c - 0.80s
3rd: Placed equations into an augmented matrix (imagine this table written as a matrix in augmented form)
4th: I reduced the matrix into row echelon form
5th: After seeing the matrix, I noticed that there was a free variable. I made s as the free variable and defined it as t because I wanted to make the equation into vertex form. (s=t)
c-s = 0 --> c-t = 0
6th: From there, I found that c equals t. So this means that c also equals s.
c=t
c=s
I know that if I find c, I can find s. This is where I get stuck on. How do I incorporate $20 million to find c?
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Wait, now that I think about it, is the answer $10 million because both coal c and steel s are equal to each other. Both c and s need to add up to $20 million.
Is this correct?
Know:
- coal and steel industries form a closed economy
- Coal Industry:
- For every $1 coal produced, $0.30 of coal needed and $0.70 of steel needed
- Steel Industry:
- For every $1 steel produced, $0.80 of coal needed and $0.20 of steel needed
1st: I created a table to help me understand this problem more
| c | s | Total Produced | |
| coal, c | $0.30 | $0.70 | $1 |
| steel, s | $0.80 | $0.70 | $1 |
2nd: I wrote 2 systems of equations and then rewrote them into homogeneous linear system of equations
c= 0.30c + 0.70s --> 0= -0.70c + 0.70s
s= 0.80c + 0.20s --> 0= 0.80c - 0.80s
3rd: Placed equations into an augmented matrix (imagine this table written as a matrix in augmented form)
| -0.70 | 0.70 | 0 |
| 0.80 | -0.80 | 0 |
4th: I reduced the matrix into row echelon form
| 1 | -1 | 0 |
| 0 | 0 | 0 |
5th: After seeing the matrix, I noticed that there was a free variable. I made s as the free variable and defined it as t because I wanted to make the equation into vertex form. (s=t)
c-s = 0 --> c-t = 0
6th: From there, I found that c equals t. So this means that c also equals s.
c=t
c=s
I know that if I find c, I can find s. This is where I get stuck on. How do I incorporate $20 million to find c?
-------
Wait, now that I think about it, is the answer $10 million because both coal c and steel s are equal to each other. Both c and s need to add up to $20 million.
Is this correct?
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