Proving Linearity: x(t) -> y(t)

[tronxo]

how can i prove mathematically if a system is linear or not? i mean, i know the system must obey proportionally law and superpositon, but i don't know how apply into it.
well, if anyone could help me, the systems i need to prove are:
x(t) -> y(t)= Cx(t) + k
x(t) -> y(t)= ∫ (from minus infinite to "t") x(e)d(e); where "e" is a dummy variable

 

[ratn_kumbh]

look if i/p x(t) is scaled by a1, then
o/p is y1(t)=a1*Cx(t)+k

if scaled by a2 it becomes
y2(t)=a2*Cx(t)+k

now to obey linearity if i/p is a1*x(t)+a2*x(t), then o/p should be
y(t)=y1(t)+y2(t)

but in this case if i/p is a1*x(t)+a2*x(t)
y(t)=a1*Cx(t)+a2*Cx(t)+k

which is not equal to
y1(t)+y2(t)= a1*Cx(t)+a2*Cx(t)+2k

So this sys is not linear

 

[ratn_kumbh]

Sorry made mistake

look if i/p x(t) is scaled by a1, then
o/p is y1(t)=a1*Cx(t)+k

if scaled by a2 it becomes
y2(t)=a2*Cx(t)+k

now to obey linearity if i/p is a1*x(t)+a2*x(t), then o/p should be
y(t)=y1(t)+y2(t)

but in this case if i/p is a1*x(t)+a2*x(t)
y(t)=a1*Cx(t)+a2*Cx(t)+k

which is not equal to
y1(t)+y2(t)= a1*Cx(t)+a2*Cx(t)+2k

So this sys is not linear

Sorry made a mistake there it should have been

if i/p is x1(t) , then
o/p is y1(t)=Cx1(t)+k

if i/p is x2(t) it becomes
y2(t)=Cx2(t)+k

now to obey linearity if i/p is a1*x(t)+a2*x(t), then o/p should be
y(t)=a1*y1(t)+a2*y2(t)

but in this case if i/p is a1*x(t)+a2*x(t)
y(t)=a1*Cx(t)+a2*Cx(t)+k

which is not equal to
a1*y1(t)+a2*y2(t)= a1*Cx(t)+a2*Cx(t)+(a1+a2)k

So this sys is not linear

 

[tronxo]

Thank u, I finally got the right result...