1. **State the problem:** We are given that line $\ell$ bisects segment $KP$ at point $L$. This means $L$ is the midpoint of $KP$, so $KL = LP$. 2. **Given expressions:** $KL = 7x + 1$ $LP = 8x - 3$ 3. **Set up the equation using the midpoint property:** Since $L$ bisects $KP$, $$7x + 1 = 8x - 3$$ 4. **Solve for $x$:** $$7x + 1 = 8x - 3$$ Subtract $7x$ from both sides: $$\cancel{7x} + 1 = \cancel{7x} + 8x - 3 \Rightarrow 1 = x - 3$$ Add 3 to both sides: $$1 + 3 = x - 3 + 3 \Rightarrow 4 = x$$ 5. **Find $KL$ by substituting $x=4$:** $$KL = 7(4) + 1 = 28 + 1 = 29$$ 6. **Since $KL = LP$, find $LP$:** $$LP = 8(4) - 3 = 32 - 3 = 29$$ 7. **Find $KP$ by adding $KL + LP$:** $$KP = KL + LP = 29 + 29 = 58$$ **Final answer:** $$\boxed{58}$$