Testing relationship between three variables

I've been asked to analyse some data relating to care outcomes, and am having difficulty deciding which test to use. The data are:
- The rate of people being admitted to nursing care, per 100,000, for around 150 organisations
- The percentage of people still at home a month after being released from hospital, for the same 150 organisations
- Expenditure on care for the same 150 organisations

Initially I was just going to look at the relationship between the first two, for which I'm assuming a Pearson's correlation would be appropriate. However, it's bringing in the third variable that is complicating things for me. Is anyone able to advise on the best test to use for this?

Thanks in advance for your help.
No, people aren't exclusively tied to one - in reality most will only be use one but I couldn't be certain that in some circumstances people won't use more.
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