1. **State the problem:** We are given vectors $v = [1, 3]$ and $w = [-2, 5]$ in $\mathbb{R}^2$. We want to find scalars $r$ and $s$ such that $$r v + s w = [-1, 19].$$ 2. **Set up the equation:** Writing the vector equation component-wise, we have: $$r \cdot 1 + s \cdot (-2) = -1$$ $$r \cdot 3 + s \cdot 5 = 19$$ 3. **Write the system of linear equations:** $$\begin{cases} r - 2s = -1 \\ 3r + 5s = 19 \end{cases}$$ 4. **Solve the system:** From the first equation, express $r$ in terms of $s$: $$r = -1 + 2s$$ Substitute into the second equation: $$3(-1 + 2s) + 5s = 19$$ $$-3 + 6s + 5s = 19$$ $$11s - 3 = 19$$ $$11s = 22$$ $$s = 2$$ 5. **Find $r$:** Substitute $s=2$ back into $r = -1 + 2s$: $$r = -1 + 2 \times 2 = -1 + 4 = 3$$ 6. **Check the solution:** $$r v + s w = 3[1,3] + 2[-2,5] = [3,9] + [-4,10] = [-1,19]$$ This matches the target vector. **Final answer:** $$r = 3, \quad s = 2.$$