Solution of homogeneous question

[nhrock3]

x and y are solutions to an equation system

i know that if x and y are solutions to a homogeneous system then for any a b
ax+by (linear diversity) is also a solution

but does it go the other way around to

if x and y are solutions and ax+by also then its a homogeneus system
?

 

[fzero]

You can check this yourself. Assuming that the system is linear, then we can write

$$\hat{A} x = \alpha ,$$

where $$\hat{A}$$ is a linear operator and $$\alpha$$ is a number, not necessarily zero. There is a similar equation for y, since it's also a solution. Now operate on the linear combination and determine if your question is true.