Anorexia Nervosa or Eating Disorder: The combination of chronic back pain, multiple joint pain, tall and thin stature, low BMI, and tachycardia in a 16-year-old female suggests a possible eating disorder, particularly anorexia nervosa. The low BMI and tall, thin build are classic physical signs, while the chronic back pain and joint pain could be related to osteoporosis or osteopenia secondary to malnutrition.

Osteoporosis: Given the patient's low BMI and tall, thin stature, osteoporosis is a possible diagnosis, especially if the patient has a history of inadequate nutrition or eating disorders. This could explain the chronic back pain and joint pain.
Hyperthyroidism: Tachycardia is a common symptom of hyperthyroidism, and this condition can also lead to muscle and joint pain, as well as osteoporosis if left untreated.
Fibromyalgia: This chronic condition is characterized by widespread musculoskeletal pain, fatigue, and tender points, which could fit the patient's symptoms of chronic back pain and multiple joint pain.

Osteogenesis Imperfecta: Although rare, this genetic disorder leads to fragile bones and could explain the easy fracturing. It's crucial to consider this diagnosis to prevent further fractures and manage the condition appropriately.
Cancer (e.g., Leukemia, Lymphoma): While less likely, cancer can cause bone pain, joint pain, and systemic symptoms like tachycardia. It's essential to rule out malignancy, especially in a patient with unexplained weight loss or persistent symptoms.
Infectious Diseases (e.g., Tuberculosis, Osteomyelitis): Chronic infections can cause persistent pain, fever, and other systemic symptoms. Although less common, these diagnoses should be considered, especially if the patient has been exposed to infectious agents or has a compromised immune system.

Ehlers-Danlos Syndrome: This group of genetic disorders affects the body's connective tissue and can lead to joint hypermobility, skin hyperextensibility, and tissue fragility. While rare, it could explain some of the patient's symptoms, such as easy bruising or joint pain.
Marfan Syndrome: Another genetic disorder that affects the body's connective tissue, Marfan syndrome can cause tall stature, joint pain, and an increased risk of osteoporosis. It's a rare condition but should be considered in the differential diagnosis due to the patient's tall and thin build.