Final answer:
To represent the new price during the 'no application fee' special, the dance studio's cost equation would change from C = 50x + 75 to C = 50x by removing the $75 application fee.
Step-by-step explanation:
The dance studio's original cost equation is C = 50x + 75, where C represents the total cost and x represents the number of months of membership. The $75 in the equation represents the application fee. During the "no application fee" special in January, this fee is waived, so the new equation would only include the monthly membership cost, which is $50 per month. The new equation would be C = 50x, removing the application fee entirely from the cost calculation.