Q:

Potting soil problem.

Consider a company that specializes in potting soil. Each bag of potting soil for seedlings requires 2 units of sand, 1 unit of loam, and 1 unit of peat moss. Each bag of potting soil for general potting requires 1 unit of sand, 2 units of loam, and 1 unit of peat moss. Each bag of potting soil for hardwood plants requires 2 units of sand, 2 units of loam, and 2 units of peat moss. Find the numbers of bags of the three types of potting soil that the company can produce with the given amounts of raw materials.
They have a total of 1000 units of sand, 800 units of loam, and 700 units of peat moss.

I have absolutely no idea how to go about answering this problem. Could someone please give me some pointer? I would really appreciate it. 
Thanks!
Marc

A:

Let S be the bags of potting soil for seedlings, G be the potting soil for general potting, and H be the potting soil for hardwood  plants.
Each bag of seedling soil needs $2$ units of sand, each bag of general soil needs $1$ unit of sand, and each bag of hardwood soil needs $2$ units of sand. We have $1000$ units of sand.
We can  say $2S+G+2H=1000$. -- (1)
By similar logic for units of loam and peat moss, we get the equations
$S+2G+2H=800$. -- (2)
$S+G+2H=700$. -- (3)
By subtracting (3) from (2), we get $G=100$.
This means that there are $100$ bags of general soil produced.
We plug this in to get new equations.
$2S+2H=900$ (1)
$S+2H=600$ (2)
$S+2H=600$ (3)
Now, we subtract (2) from (1), we get $S=300$
This means that there are $300$ bags of seedling soil produced.
We plug this into equation (2)
$300 + 2H = 600$.
Therefore $H=150$, so there are $150$ bags of hardwood soil produced.
Finally, we have $100$ bags of general soil, $300$ bags of seedling soil, and $150$ bags of hardwood soil.
$\boxed{(S,G,H)=(300,100,150)}$

Notice: This solution seems complicated, but if you practice a few problems and come back to this, this entire process should take you no more than $15$ seconds, I ASSURE you.

