https://file.aiquickdraw.com/mathbot/file/1727212213060cw14hdqp.jpg

Let's work through each question one by one from the image:

1. **If \( LM = 22 \) and \( MN = 15 \), find \( LN \):**

   Since \( LN \) is the sum of the distances \( LM \) and \( MN \):
   \[
   LN = LM + MN = 22 + 15 = 37
   \]
   
   **Answer: \( LN = 37 \)**

2. **If \( LN = 54 \) and \( LM = 31 \), find \( MN \):**

   To find \( MN \), subtract \( LM \) from \( LN \):
   \[
   MN = LN - LM = 54 - 31 = 23
   \]
   
   **Answer: \( MN = 23 \)**

3. **If \( RT = 36 \), find the value of \( x \) (where \( RS = 6x + 1 \) and \( ST = x + 7 \)):**

   Since \( RT = RS + ST \), we have the equation:
   \[
   6x + 1 + x + 7 = 36
   \]
   Simplify:
   \[
   7x + 8 = 36
   \]
   Subtract 8 from both sides:
   \[
   7x = 28
   \]
   Divide by 7:
   \[
   x = 4
   \]
   
   **Answer: \( x = 4 \)**

4. **If \( DF = 9x - 39 \), find \( EF \) (where \( DE = 47 \) and \( EF = 3x + 10 \)):**

   Since \( DF = DE + EF \), we have the equation:
   \[
   47 + (3x + 10) = 9x - 39
   \]
   Simplify:
   \[
   57 + 3x = 9x - 39
   \]
   Subtract \( 3x \) from both sides:
   \[
   57 = 6x - 39
   \]
   Add 39 to both sides:
   \[
   96 = 6x
   \]
   Divide by 6:
   \[
   x = 16
   \]
   Now substitute \( x = 16 \) into \( EF = 3x + 10 \):
   \[
   EF = 3(16) + 10 = 48 + 10 = 58
   \]
   
   **Answer: \( EF = 58 \)**

5. **If \( UW = 6x - 35 \), find \( UW \) (where \( UV = 19 \) and \( VW = 4x - 20 \)):**

   Since \( UW = UV + VW \), we have:
   \[
   19 + (4x - 20) = 6x - 35
   \]
   Simplify:
   \[
   19 + 4x - 20 = 6x - 35
   \]
   Combine like terms:
   \[
   -1 + 4x = 6x - 35
   \]
   Subtract \( 4x \) from both sides:
   \[
   -1 = 2x - 35
   \]
   Add 35 to both sides:
   \[
   34 = 2x
   \]
   Divide by 2:
   \[
   x = 17
   \]
   Now substitute \( x = 17 \) into \( UW = 6x - 35 \):
   \[
   UW = 6(17) - 35 = 102 - 35 = 67
   \]
   
   **Answer: \( UW = 67 \)**

6. **If \( HJ = 7x - 27 \), find the value of \( x \) (where \( HI = 3x - 5 \) and \( IJ = x - 1 \)):**

   Since \( HJ = HI + IJ \), we have the equation:
   \[
   (3x - 5) + (x - 1) = 7x - 27
   \]
   Simplify:
   \[
   4x - 6 = 7x - 27
   \]
   Subtract \( 4x \) from both sides:
   \[
   -6 = 3x - 27
   \]
   Add 27 to both sides:
   \[
   21 = 3x
   \]
   Divide by 3:
   \[
   x = 7
   \]
   
   **Answer: \( x = 7 \)**

7. **If \( BD = 7x - 10 \), \( BC = 4x - 29 \), and \( CD = 5x - 9 \), find each value:**

   Since \( BD = BC + CD \), we have the equation:
   \[
   (4x - 29) + (5x - 9) = 7x - 10
   \]
   Simplify:
   \[
   9x - 38 = 7x - 10
   \]
   Subtract \( 7x \) from both sides:
   \[
   2x - 38 = -10
   \]
   Add 38 to both sides:
   \[
   2x = 28
   \]
   Divide by 2:
   \[
   x = 14
   \]
   Now substitute \( x = 14 \) into each equation:

   - \( BD = 7(14) - 10 = 98 - 10 = 88 \)
   - \( BC = 4(14) - 29 = 56 - 29 = 27 \)
   - \( CD = 5(14) - 9 = 70 - 9 = 61 \)
   
   **Answers:**
   - \( BD = 88 \)
   - \( BC = 27 \)
   - \( CD = 61 \)

Would you like to dive into any specific part of the solution in more detail? Here are some related questions:

1. How can you solve similar problems involving linear segments?
2. How do you set up and simplify linear equations?
3. How does combining like terms help solve algebraic expressions?
4. Can you explain how to handle equations where multiple segments are added?
5. How does substituting values back into the equation help verify the solution?

**Tip:** When solving equations, always ensure the expressions on both sides of the equation are simplified before solving for the variable.

Solve the set of problems involving linear segments, algebraic equations, and their corresponding values, as shown in the diagram.

Learn how to solve algebraic problems involving segment addition and the solving of linear equations. This page offers a detailed step-by-step explanation, covering concepts like segment addition, simplification, and substitution for students in Grades 8-10.

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