This pattern expresses ability or possibility.

When someone or something is able to do something, or when something is possible, -(으)ㄹ 수 있다 is used, and when someone or something is not able to do something, or when something is not possible, -(으)ㄹ 수 없다 is used.

In English it means 'can'. When a verb stem ends in a vowel or ㄹ, -ㄹ 수 있다/없다 is used, and when a verb stem ends in a consonant, -을 수 있다/없다 is used.

Details

Adding the particle -가 to -(으)ㄹ 수 있다/없다 to form -(으)ㄹ 수가 있다/없다 makes the phrase more emphatic in meaning than -(으)ㄹ 수 있다/없다 alone.

떡볶이가 매워서 먹을 수 없어요.

The ddeokbokki is spicy, so I can't eat it.

맥볶이가 매워서 먹을 수가 없어요.

The ddeokbokki is so spicy that I (really) can't eat it.

길이 막혀서 갈 수 없어요.

There's a lot of traffic, so I won't be able to make it.

길이 막혀서 갈 수가 없어요.

There's so much traffic that I really can't make it no matter what.

Kimchi Reader

A pattern to expresses when someone is able to do something or when something is possible.

Details

Emphasis with 수가

The particle 가 can be added to 수 to make the sentence more emphatic.

그렇게 사랑했던 두 사람이 헤어지다니 믿을 수가 없어요.

I can't believe that the two people who loved each other so much broke up.

V-(으)ㄹ 수 있다/없다

Kimchi Reader

Emphasis with 수가