1. **State the problem:** We need to find the value of $y$ for the point $P = \left(-\frac{1}{7}, y\right)$ that lies on the unit circle centered at the origin. 2. **Recall the equation of a unit circle:** The unit circle centered at the origin has the equation $$x^2 + y^2 = 1$$ where the radius is 1. 3. **Substitute the given $x$-coordinate:** Plug in $x = -\frac{1}{7}$ into the equation: $$\left(-\frac{1}{7}\right)^2 + y^2 = 1$$ 4. **Simplify the expression:** $$\frac{1}{49} + y^2 = 1$$ 5. **Isolate $y^2$:** $$y^2 = 1 - \frac{1}{49} = \frac{49}{49} - \frac{1}{49} = \frac{48}{49}$$ 6. **Solve for $y$:** $$y = \pm \sqrt{\frac{48}{49}} = \pm \frac{\sqrt{48}}{7}$$ 7. **Simplify the square root:** $$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$$ 8. **Final simplified form:** $$y = \pm \frac{4\sqrt{3}}{7}$$ **Answer:** The value of $y$ in simplest form is $$\pm \frac{4\sqrt{3}}{7}$$.