1. **State the problem:** Given that triangles $\triangle JKL$ and $\triangle NMP$ are similar, find the value of $x$. 2. **Recall the property of similar triangles:** Corresponding sides of similar triangles are proportional. This means: $$\frac{JK}{NM} = \frac{JL}{NP} = \frac{KL}{MP}$$ 3. **Identify corresponding sides:** - $JK$ corresponds to $NM$ - $JL$ corresponds to $NP$ - $KL$ corresponds to $MP$ 4. **Substitute the given values:** - $JK = 49$ - $JL = 9x + 1$ - $NM = 14$ - $NP = x + 5$ 5. **Set up the proportion using the corresponding sides:** $$\frac{JK}{NM} = \frac{JL}{NP}$$ $$\frac{49}{14} = \frac{9x + 1}{x + 5}$$ 6. **Simplify the left side:** $$\frac{49}{14} = \cancel{\frac{7 \times 7}{7 \times 2}} = \frac{7}{2}$$ 7. **Write the equation:** $$\frac{7}{2} = \frac{9x + 1}{x + 5}$$ 8. **Cross multiply:** $$7(x + 5) = 2(9x + 1)$$ 9. **Expand both sides:** $$7x + 35 = 18x + 2$$ 10. **Bring all terms to one side:** $$7x + 35 - 18x - 2 = 0$$ $$-11x + 33 = 0$$ 11. **Solve for $x$:** $$-11x = -33$$ $$x = \frac{-33}{-11}$$ $$x = \cancel{\frac{-33}{-11}} = 3$$ **Final answer:** $$\boxed{3}$$