Answer
The amounts invested (in dollars) are: $4000$ ($10\%$), $8000$ ($12\%$), $5000$ ($15\%$),
Work Step by Step
Step 1. Assume the amounts invested (in dollars) are: $x$ ($10\%$), $y$ ($12\%$), $z$ ($15\%$),
Step 2. Based on the given conditions, we have
$\begin{cases} x+y+z=17,000\\ 0.1x+0.12y+0.15z=2110 \\ y=x+z-1000 \end{cases}$
Step 3. Simplify and rearrange the above to get
$\begin{cases} x+y+z=17,000\\ 10x+12y+15z=211,000 \\ x-y+z=1000 \end{cases}$
Step 4. Eliminating $x+z$ using equations 1 and 3, we have $2y=16,000$; thus $y=8000$
Step 5. Using the y-value, we get $\begin{cases} x+z=9000\\ 10x+15z=115,000 \end{cases}$
Step 6. Multiply 10 to the first equation and take the difference from the second; we get $5z=25,000$. Thus $z=5000$ and $x=4000$
Step 7. We conclude that the amounts invested (in dollars) are: $4000$ ($10\%$), $8000$ ($12\%$), $5000$ ($15\%$),
