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No, not all do. The proportionality constants that change the units will have units themselves.
All numbers are constants.
Variables change, constants do not.
An equation is linear when it contains only variables of degree 1 and constants. ALL linear equations will be of the form: a1x1+a2x2+a3x3+...+anxn=c where an and c are constants.
Constants in the sense you mean are different to mathematical constants. They usually refer to a system in which two values are proportionate and so are of the units associated with the system.