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

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Borys Shumyatskiy eNotes educator| Certified Educator


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.