Question: 3. If $RT = 36$, find the value of $x$.
Given segments: $RS = 6x + 1$, $ST = x + 7$
5. If $UW = 6x - 35$, find $UW$.
Given segments: $UV = 19$, $VW = 4x - 20$
1. **Problem 3:** Find $x$ given $RT = 36$, $RS = 6x + 1$, and $ST = x + 7$.
2. Since $R$, $S$, and $T$ are collinear and $RT$ is the whole segment, we use the segment addition postulate:
$$RT = RS + ST$$
3. Substitute the given expressions:
$$36 = (6x + 1) + (x + 7)$$
4. Simplify the right side:
$$36 = 6x + 1 + x + 7$$
$$36 = 7x + 8$$
5. Subtract 8 from both sides:
$$36 - 8 = 7x + \cancel{8} - \cancel{8}$$
$$28 = 7x$$
6. Divide both sides by 7:
$$\frac{28}{7} = \frac{7x}{7}$$
$$4 = x$$
7. **Answer for problem 3:** $x = 4$
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8. **Problem 5:** Find $UW$ given $UW = 6x - 35$, $UV = 19$, and $VW = 4x - 20$.
9. Since $U$, $V$, and $W$ are collinear and $UW$ is the whole segment, use the segment addition postulate:
$$UW = UV + VW$$
10. Substitute the given expressions:
$$6x - 35 = 19 + (4x - 20)$$
11. Simplify the right side:
$$6x - 35 = 19 + 4x - 20$$
$$6x - 35 = 4x - 1$$
12. Subtract $4x$ from both sides:
$$6x - 4x - 35 = 4x - 4x - 1$$
$$2x - 35 = -1$$
13. Add 35 to both sides:
$$2x - 35 + 35 = -1 + 35$$
$$2x = 34$$
14. Divide both sides by 2:
$$\frac{2x}{2} = \frac{34}{2}$$
$$x = 17$$
15. Find $UW$ by substituting $x=17$ into $6x - 35$:
$$UW = 6(17) - 35 = 102 - 35 = 67$$
16. **Answer for problem 5:** $UW = 67$