Hi, I am working with a database containing school applications of students during year 2019 (rank-ordered applications, but for simplicity, lets assume that I can only distinguish between the most preferred school and the remaining schools), home-school distance, academic quality of schools (avg. std test scores), tuition fees charged by each school, and the share of vulnerable Students at each school (low_SES_share). The goal is to identify school preferences of students/parents.

I am estimating a linear probability model Y = b0 + b1Distance + b2AcademicQuality + b3TuitionFees + b4Low_SES_share, where Y=1 for the preferred school, Y=0 otherwise (I also include interactions for vulnerable students, i.e., low-SES students, but that's not relevant here).

Low_SES_share is equivalent to 1-High_SES_share. Till now, I thought that considering Low-SES-share or High-SES-share (=1-Low-SES-share) in the equation would deliver the same results, except for b4, the coefficient related to that variable. However, if rather than considering low-SES-share I consider High-SES-share, not only the estimated coefficient b4 changes but also the other ones (related to tuition fees, distance and academic quality).
How does that make sense? If I'm just introducing a linear transformation of a variable, why do the other coefficients also change?