Subject: Apology for Miscommunication
Dear [Recipient's Name],
I hope this message finds you well. I want to sincerely apologize for the recent misunderstanding regarding the equation we discussed. It seems I mistakenly used the value of x instead of replacing it with 5 when solving for y.
I appreciate your patience and understanding as I've worked through this error. Your insights and guidance have always been helpful, and I'm grateful for your support.
Thank you for your understanding, and I assure you that I will double-check my work in the future to avoid similar mistakes.
Best regards,
[Your Name]