Question

Find m and b so that y=mx+b is a solution to the differential equation
dy/dx=1/2x+y-1

Answer 1

I will assume you mean (1/2)x and not 1/(2x)in the differential equation

dy/dx = m, so we require that
m = x/2 + mx+ b -1

This is true if m = -1/2 and
b = 1+m = 1/2

y = -x/2 + 1/2 is a solution, but not the only solution, to the differential equation.

Answer 2

dy/dx=(1/2)x + y - 1
I assume that is what you mean.
y = m x + b
dy/dx = m
so
m = .5 x + mx + b - 1
m = (.5 + m) x + b-1
well if m is to be constant, m must be -.5
that means
-.5 = b-1
b = .5
so
y = -.5 x + .5
--------the end -----
Now check with m = -.5 and b = +.5
y = -.5 x + .5
dy/dx = -.5

dy/dx = .5 x + (-.5 x +.5) - 1
= +.5 - 1
= -.5
check