Proving that sum of two distinct lines in $\mathbb{R}^2$ is $\mathbb{R}^2$?

Question

How do I prove that the sum of two lines, for example $y=3x$ and $y=8x$, is $\mathbb{R}^2$? Is there a way I can show that every point in $\mathbb{R}^2$ can be written as a sum of these two lines?

Answer 1

Points on $y=3x$ have the form $(r,3r)$.

Points on $y=8x$ have the form $(s,8s)$.

Points on the sum of these two lines have the form $(r,3r)+(s,8s)=(r+s,3r+8s)$.

Given any point $(a,b) \in \mathbb{R}^2$, set $(a,b)=(r+s,3r+8s)$ and solve the resulting system of equations for $r,s$: $$r+s=a$$ $$3r+8s=b$$ .

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Answer 2

Lines through origin in $\Bbb R^2$ are one-dimensional subspaces. Thus, given two distinct lines through origin, you get subspaces $L$ and $K$ of $\Bbb R^2$ such that $L\cap K=\{0\}$. This means that sum $L+K$ is direct and $$\dim(L+K) = \dim L + \dim K = 2\implies L + K = \Bbb R^2$$