1. **State the problem:** We are given a system of linear equations: $$2x + 3y = 1$$ $$7x + 10y = 5$$ We need to find the value of $$10x_1 + y_1$$ where $$(x_1, y_1)$$ is the solution to the system. 2. **Use the substitution or elimination method:** Here, we use elimination. 3. Multiply the first equation by 7 and the second by 2 to align coefficients of $x$: $$7(2x + 3y) = 7(1) \Rightarrow 14x + 21y = 7$$ $$2(7x + 10y) = 2(5) \Rightarrow 14x + 20y = 10$$ 4. Subtract the second from the first to eliminate $x$: $$ (14x + 21y) - (14x + 20y) = 7 - 10 $$ $$ 14x - \cancel{14x} + 21y - 20y = -3 $$ $$ y = -3 $$ 5. Substitute $y = -3$ into the first original equation: $$ 2x + 3(-3) = 1 $$ $$ 2x - 9 = 1 $$ $$ 2x = 10 $$ $$ x = \frac{10}{2} = 5 $$ 6. Now compute $10x_1 + y_1$: $$ 10(5) + (-3) = 50 - 3 = 47 $$ **Final answer:** $$\boxed{47}$$