is the sum of two linear expressions always a linear expression?

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In general, the answer is "yes". Unfortunately your question isn't very specific, so I'll try to guess.

If you mean linear functions of one variable, they have the form y=ax+b. If another function is y=cx+d, then their sum is

y=(a+c)x + (b+d),

and it also has this form, some constant multiplied by x plus one more constant.

Moreover, if we multiply linear function by some constant, the resulting function remains linear:

r*(ax+b) = (ra)x + (rb).

This is true for linear functions of two or more variables:

(a1*x1 + b1*x2 + c1) + (a2*x1 + b2*x2 + c2) = (a1+a2)*x1 + (b1+b2)*x2 + (c1+c2).

There is also a large and beautiful theory of general linear spaces (even infinite dimensional), in it linearity of operator T means by definition that T(a*x+b*y)=aT(x)+bT(y) for any numbers a, b and any "vectors" x, y.

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