The terms to fill the blanks are: tienes, tengo, tenemos, tiene, tienes, tengo.
So the sentences will be:
ELENA: Oye, Carlos! ¿Cuántos años tienes?
CARLOS: Yo? Yo tengo quince años. ¿Por qué?
ELENA: Porque mi hermano y yo tenemos una prima de quince años que tiene que ir a un baile el viernes. (Tú) ¿tienes planes?
CARLOS: El viernes? No, no tengo otros planes.
In the first sentence, the correct form is taken because it was directed to Carlos as a second person of the singular.
In the second speech, the correct form is tengo because it is available to you in the first person of the individual.
In the third sentence, the correct form is tenemos because Elena and her husband are the first person of the plural. In the second part, the correct form is due to the fact that it was directed to Carlos as a second person of the singular.
See full text below
Fill in each blank with the correct conjugation of the verb tener.
ELENA: Oye, Carlos! ¿Cuántos años ___________?
CARLOS:Yo? Yo ___________ quince años. ¿Por qué?
ELENA: Porque mi hermano y yo __________ una prima de quince amos que _____________ que ir a un baile el viernes. (Tu) _____________ planes?
CARLOS:El viernes? No, no ____________ otros planes.