Ive been trying to find the correct system of equation for this one problem for a long time and since i dont want to disturb anyone else in the office with my schoolwork Im posting it here where I could find someno:
There is a coffee merchant who is selling coffee beans. 500g of type a cost €8.-, 500g of type b €10.- and 500g of type c €12.- .
a) A mixture of 500g of each of these types costs €9.- .
Establish an equation for the amount of each type in g and state every solution (a,b,c ∈]0 ; 1[)
You should differentiate between weight ($m$) and price ($p$). In this problem the prices are given (as per gram), but the weights in the questioned mixture are unknown. You can approach it in the following way.
The costs ($\$$ per gram) are \begin{align}p_{\mathrm{a}}&=\frac{8}{500}\,(\$/\mathrm{g})\\p_{\mathrm{b}}&=\frac{10}{500}\,(\$/\mathrm{g})\\p_{\mathrm{c}}&=\frac{12}{500}\,(\$/\mathrm{g})\end{align}
Now, we know some $500\,\mathrm{g}$ mixture of the described coffee types costs $9\,\$$. Thus, assuming $m_{\mathrm{a}}$, $m_{\mathrm{b}}$ and $m_{\mathrm{c}}$ to be the unknown weights of the coffee types in the mixture, we have $$m_{\mathrm{a}}+m_{\mathrm{b}}+m_{\mathrm{c}}=500\,(\mathrm{g})$$ and since the price of the mixture is given $9\,\$$ we can write $$p_{\mathrm{a}}m_{\mathrm{a}}+p_{\mathrm{b}}m_{\mathrm{b}}+p_{\mathrm{c}}m_{\mathrm{c}}=9\,\$$$
Unfortunately, to solve the problem you'll need one more equation since there are three unknowns $m_{\mathrm{a}}$, $m_{\mathrm{b}}$ and $m_{\mathrm{c}}$ and still two equations.