"My older sister is busy."
Best to think of 很 as "is" or can think in Chinese is something isn't spoken about as "very" then it isn't worth mentioning :)
Eg. 我很忙 , I'm (very) busy 我很好, I'm (very) good 我的家很大, My home is (very) big
I'm sure more words for 'very' will be taught later that actually mean 'very'.
Akavel is more right than BriaunnaYo here. You often need to use 很 for balance in a sentence instead of 是 though it behaves the same way in a lot of situations. For a simple example : ”我很好“，意思是 "I'm good", not "I'm very good." But literally translated it's more like "I very good" The 很 acts like the 'to be' verb. You wouldn't say "我是好“ as it would be wrong. But actually this is the problem with literal translations... because Chinese skips the "being" verb a lot when it is using an adjective to describe the way that something is being. After all, if you're saying that something is a certain way, there's really no need to say that it "is" because isn't that implied by the fact that it can be described in any way? 比如说。。。 “我很累”，"今天天气太热", "那个女士非常漂亮".
If anyone sees any mistake I've made let me know... this is kind of a weird concept to explain.
If you want to stress the 很 (very), you'd need to modify the sentence to something like: 我真的很忙。(I'm truly very busy). Adding another adverb in there changes the nature of "很" from adverb that links the subject to the adjective to an adverb that helps describe the verb.