In a music club with 23 members, 10 people played the piano, 9 people played the guitar, and 5 people didn't play either of these two instruments. How many club members played both the piano and the guitar?

1 member played both instruments.

Step-by-step explanation:

Lets call:

A: the number of people that only play the piano

B: the number of people that played both

C: the number of people that only play the guitar

D: the number of people that didn't play either of these two instruments.

If the music club has 23 members, we can write the following equation:

A + B + C + D = 23 (Equation 1)

At the same way, from the sentences 10 people played the piano, 9 people played the guitar, and 5 people didn't play either of these two instruments, we can write the following equations:

A + B = 10 (Equation 2)

B + C = 9 (Equation 3)

D = 5 (Equation 4)

So, replacing equation 2 and 4 on equation 1 and solving for C, we get:

(A+B) + C + D = 23

10 + C + 5 = 23

C + 15 = 23

C = 23 - 15

C = 8

Then, replacing the value of C on equation 2 and solving for B, we get:

B + C = 9

B + 8 = 9

B = 9 - 8

B = 1

Finally the number of club members that played both the piano and the guitar is 1 person.