1. **State the problem:** We are given a line with equation $y - 2x = 7$ which is the perpendicular bisector of the line segment $AB$ where $A = (j, 7)$ and $B = (6, k)$. We need to find the coordinates of the midpoint of $AB$. 2. **Rewrite the line equation:** The line equation can be rewritten as $$y = 2x + 7$$ 3. **Recall properties of perpendicular bisector:** - The perpendicular bisector passes through the midpoint of $AB$. - The slope of the perpendicular bisector is the negative reciprocal of the slope of $AB$. 4. **Find midpoint coordinates:** The midpoint $M$ of $AB$ is $$M = \left(\frac{j + 6}{2}, \frac{7 + k}{2}\right)$$ 5. **Since $M$ lies on the perpendicular bisector, substitute $M$ into the line equation:** $$\frac{7 + k}{2} = 2 \cdot \frac{j + 6}{2} + 7$$ 6. **Simplify the right side:** $$\frac{7 + k}{2} = (j + 6) + 7 = j + 13$$ 7. **Multiply both sides by 2 to clear the denominator:** $$7 + k = 2j + 26$$ 8. **Rearranged:** $$k = 2j + 19$$ 9. **Find slope of $AB$:** $$m_{AB} = \frac{k - 7}{6 - j}$$ 10. **Slope of perpendicular bisector is 2 (from $y=2x+7$). Since it is perpendicular to $AB$, slopes satisfy:** $$m_{AB} \times 2 = -1 \implies m_{AB} = -\frac{1}{2}$$ 11. **Set slope of $AB$ equal to $-\frac{1}{2}$:** $$\frac{k - 7}{6 - j} = -\frac{1}{2}$$ 12. **Cross multiply:** $$(k - 7) \times 2 = -1 \times (6 - j)$$ $$2k - 14 = -6 + j$$ 13. **Rearranged:** $$2k - j = 8$$ 14. **Substitute $k = 2j + 19$ from step 8 into this equation:** $$2(2j + 19) - j = 8$$ $$4j + 38 - j = 8$$ $$3j + 38 = 8$$ 15. **Solve for $j$:** $$3j = 8 - 38 = -30$$ $$j = -10$$ 16. **Find $k$ using $k = 2j + 19$:** $$k = 2(-10) + 19 = -20 + 19 = -1$$ 17. **Find midpoint coordinates:** $$M = \left(\frac{j + 6}{2}, \frac{7 + k}{2}\right) = \left(\frac{-10 + 6}{2}, \frac{7 + (-1)}{2}\right) = \left(\frac{-4}{2}, \frac{6}{2}\right) = (-2, 3)$$ **Final answer:** The midpoint of $AB$ is $\boxed{(-2, 3)}$.