Off-Diagonal Continuous Rado Numbers $x_1+x_2+\cdots +x_k=x_0$

In 2001, Robertson and Schaal found the 2-color off-diagonal generalized Schur numbers: for two positive integers $k$ and $l$, they determined the smallest positive integer $S=S(k,l)$ such that for any coloring of the integers from 1 to $S$ using red and blue, there must be a red solution to the equation $x_1+x_2+\cdots +x_k=x_0$ or a blue solution to the equation $x_1+x_2+\cdots +x_l=x_0$. We extend this result to find the continuous version: for two positive integers $k$ and $l$, we find the smallest real number $S=S_{\mathbb{R}}(k,l)$ such that for any coloring of the real numbers from 1 to $S$ using red and blue, there must be a red solution to the equation $x_1+x_2+\cdots +x_k=x_0$ or a blue solution to the equation $x_1+x_2+\cdots +x_l=x_0$.