1. **State the problem:** We are given a line with slope $m=2$ and an $x$-intercept at $\left(\frac{8}{7},0\right)$. We need to find the $y$-intercept of this line. 2. **Recall the slope-intercept form of a line:** $$y = mx + b$$ where $m$ is the slope and $b$ is the $y$-intercept. 3. **Use the given $x$-intercept to find $b$:** At the $x$-intercept, $y=0$ and $x=\frac{8}{7}$. Substitute these values and $m=2$ into the equation: $$0 = 2 \times \frac{8}{7} + b$$ 4. **Solve for $b$:** $$0 = \frac{16}{7} + b$$ $$b = -\frac{16}{7}$$ 5. **Interpret the result:** The $y$-intercept is at $\left(0, -\frac{16}{7}\right)$. 6. **Check the options:** Option d. $\left(0, -\frac{16}{7}\right)$ matches our result. **Final answer:** The $y$-intercept is $\boxed{\left(0, -\frac{16}{7}\right)}$.