Answer: Unfortunately I don't know for certain which bamboo you have because I can find no reference to "American" bamboo. However, it is undoubtedly, a running-type which spreads via shallow rhizomes or underground shoots. Still the best and surest method is digging by hand. But in the best circumstances, this will take season long (even two season) diligence. You might consider hiring a landscape firm or laborers to help you schedule monthly bamboo "digs."
Cutting off all new shoots at ground level also helps by gradually depleting the roots of energy. But this will take even more time. If you try this, don't allow any shoots to exceed 2 feet in height.
Herbicides such a glyphosate (Roundup) are less effective on bamboo than most plants. Also, since you're working on a slope, you should consider what might happen if rains come post application and wash the herbicide away. Where will it collect? Before using any chemical herbicide, I recommend you check with your local cooperative extension office.
You can eliminate bamboo by digging. I suspect it will require one massive effort and follow-up with somewhat less intensive efforts.
I hope this helps.
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