Hello kathirine94,
Thanks for your post to Apple Support Communities. We understand you want to be able to send and receive SMS (green) text messages on your iPhone. We'll walk through steps that should accomplish what you'd like. First, be aware that SMS text messages are through your carrier. iMessage or MMS (blue) text messages are associated with your Apple ID and can be sent and received on your Mac once you've signed in with your Apple ID.
In order to receive SMS text messages on your MacBook Air, you will have to set it text message forwarding from your iPhone to your Mac. We'll use steps from this article: Forward SMS/MMS text messages from your iPhone to your Mac or iPad - Apple Support
On your Mac:
- Open Messages
- Choose the Messages menu > Settings > iMessage tab
- Confirm the Apple ID shown logged in is the same one on your iPhone
On your iPhone:
- Go to Settings > Messages > Text Message Forwarding
- Confirm your MacBook Air shows and is enabled
If you need to confirm you are logged in with the same Apple ID on your iPhone, go to Settings > Messages > Send & Receive. These steps should allow SMS text messages to forward from your iPhone when they come in to your MacBook Air in the Messages app. Reply back with any concerns.
Take care.