Segment Length 3702Dd

1. **State the problem:** We have a line segment $QS$ with point $R$ between $Q$ and $S$. We know the lengths $RS = 2x - 1$, $QR = 5x + 3$, and the total length $QS = 5x + 10$. We need to find the numerical length of $QS$. 2. **Use the Segment Addition Postulate:** The length of $QS$ is the sum of the lengths $QR$ and $RS$ because $R$ lies between $Q$ and $S$. $$QS = QR + RS$$ 3. **Substitute the given expressions:** $$5x + 10 = (5x + 3) + (2x - 1)$$ 4. **Simplify the right side:** $$5x + 10 = 5x + 3 + 2x - 1$$ $$5x + 10 = 7x + 2$$ 5. **Solve for $x$:** Subtract $5x$ from both sides: $$\cancel{5x} + 10 = \cancel{5x} + 7x + 2$$ $$10 = 2x + 2$$ Subtract $2$ from both sides: $$10 - 2 = 2x + \cancel{2} - \cancel{2}$$ $$8 = 2x$$ Divide both sides by $2$: $$\frac{8}{\cancel{2}} = \frac{2x}{\cancel{2}}$$ $$4 = x$$ 6. **Find the length of $QS$ by substituting $x=4$ into $QS = 5x + 10$:** $$QS = 5(4) + 10 = 20 + 10 = 30$$ **Final answer:** The length of $QS$ is $30$.