ordinary differential equations - Homogeneous First Order ODEs

I came across this problem today in Differential Equations with Applications and Historical Notes by George F. Simons, and I almost do not know how to start off. I need help. It says,

Show that the substitution
z = ax + by + c
changes
y′ = f (ax + by + c)

into an equation with separable variables.

Answer

Just substitute $z=ax+by+c \to z'=by'+a$

$$y′ = f (ax + by + c)$$
$$y' =f(z)$$
For $b \ne 0$
$$\frac {z'-a}{b}=f(z)$$
$${z'}=bf(z)+a$$
this form of the equation is separable

$$\int\frac {dz}{bf(z)+a}=\int dx=x+K$$

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Edit for $b=0$
It's already separable..
$$y′ = f (ax + c)$$
$$\int dy =\int f (ax + c)dx $$
$$y =\int f (ax + c)dx $$