A retired couple wishes to have an additional annual income of $\$6000$ per year.
As their financial consultant, you recommend that they invest some money in Treasury Bills ($t$) that yield $6$%, some in corporate paper ($p$) yielding $3$%, some in corporate bonds ($b$) that yield $4$%, and some in junk bonds ($j$) that yield $10$%. Suppose the couple have $120000 to invest. Set up and solve a system of equations for this situation. Note any natural constraints on the variables (at least four inequalities needed). Graph the domain on a labeled and scaled coordinate axis (that is graph the inequalities and shade the appropriate region). Scale the axes so that the graph is large and clear. Copy and fill in the table below to show various ways their goal can be achieved. Include one option where no category has zero invested.
What I have so far: I put the system into a matrix through the equations\begin{align*} t+p+b+j &= 120000\\ 0.06t+0.03p+0.04b+0.1j &= 6000 \end{align*}
From the ref, I got the following info:\begin{align*} t + \frac{b}{3} + \frac{7}{3}j &= 8000\\ p + \frac{2}{3}b - \frac{4}{3}j &= 40000 \end{align*} The free variables are $b$ and $j$.
Next, we have to determine the natural constraints. Obviously, two are $b,j\geqslant0$. I put down the other two as $b+j \leqslant 120000$ and $b+j \geqslant 0$.
But I think I need more constraints, since after plotting the domain of $b$ and $j$, and identifying possible combinations, I was getting totals over $120,000$, which is the maximum that the couple can spend.
The question that I have is what "natural constraint" am I missing?
