How to solve a linear-algebra word problem The question:
The cost to produce $x$ number of sport hats is $c = 200 + 25x$. The selling price is $45$. Approximately how many hats were sold if the seller made a profit of $3000$?
From the question I can derive:
$P$ (Price) $= 45\$$
$TC$ (Total Cost) $= FC$ (Fixed Cost) $+ VC$ (Variable Cost) $= 200 + 25x$ (where $x$ is in terms of quantity $Q$)
$\pi$ (Profit) $= 3000\$$
I have tried to substitution, elimination and also trying to calculate revenue to work it back to quantity.
e.g. 
$\pi = TR - TC$ ($TR = P\cdot Q$ and $TC = 200 + 25x$)
... $3000 = P\cdot Q - 200 + 25Q$ (assuming that $x$ is also quantity)
This does not calculate correctly.
If I take the substitution approach deriving liner formulas
cost $= y = 200 + 25x$ (in terms of quantity sold)
revenue $= y = 45x$ (in terms of quantity sold)
then substituting gives me
$45Q = 200 + 25x$
...$20x = 200$
...$x = 10$?
So if $x$ is $10$ it doesn't solve the quantity sold.
Where do you begin to analyse the calculate such problems.
I am not so concerned about the answer of the word problem but rather how to calculate it and future ones?
 A: I don't think this is linear algebra--it just looks like algebra to me.  The profit is the revenue minus the cost which you can immediately write down:
$$
P = 45x - (200 + 25x) = 3000 \\
20x - 200 = 3000
$$
Now solve for x.
A: You can solve this without really using any formulas, but rather by means of words and common sense. The formula for cost of production $c = 200 + 25x$ basically tells you this: first you pay \$200 to set things up. Rent the building, put up a sign at the entrance, etc. Then, after this, you pay \$25 for making each hat (these probably go to workers' salary and cost of raw materials).
Now, you've made a certain amount of hats and sold them for \$45 each. You made a profit of \$3000. Note that in the very beginning you had to pay \$200 as a set up cost. If you hadn't paid these \$200 (for instance, your friend let you use his basement and equipment for free), then your profit would have been even higher: it would have been \$3200. So, let us forget about the setup cost and think that our profit was \$3200.
Now it's simple arithmetic. We pay \$25 to produce each hat, and we sell the hat for \$45. So each hat gives us \$20 of profit. Our overall profit is \$3200, which means we must have made $3200/20 = 160$ hats. So the answer is 160.
