Let's define the number of students who signed up for music lessons last year as \( x \).
According to the problem, this year 250 students signed up for music lessons, and this number is 42 less than the number of students last year. We can express this relationship with the following equation:
\[ 250 = x - 42 \]
To find \( x \), we can solve the equation by adding 42 to both sides:
\[ 250 + 42 = x \]
\[ x = 292 \]
So, the number of students who signed up for music lessons last year is 292.