Let's solve the problem step by step:

Problem Details:

1. Line Segment LN is 40 units.
2. LM is $(3x + 2)$ units.
3. MN is $(12 + 3x)$ units.
4. You need to:
   • Solve for $x$.
   • Then find the length of LM.

Step 1: Express the relationship between the segments

The total length of LN is the sum of LM and MN: $LM + MN = LN$ Substitute the given expressions: $(3x + 2) + (12 + 3x) = 40$

Step 2: Simplify the equation

Combine like terms: $6x + 14 = 40$

Step 3: Solve for $x$

Subtract 14 from both sides: $6x = 26$ Divide by 6: $x = \frac{26}{6} = \frac{13}{3}$

Step 4: Find LM

Substitute $x = \frac{13}{3}$ into the expression for LM: $LM = 3x + 2$ $LM = 3 \times \frac{13}{3} + 2 = 13 + 2 = 15 \, \text{units}.$

Final Answer:

The length of LM is 15 units.

Would you like further clarification or explanation?

Here are 5 questions to expand your understanding:

1. How would the solution change if the total length of LN were different?
2. What if LM and MN had expressions involving different variables?
3. How do we check the accuracy of our solution?
4. Why is substituting back the value of $x$ important?
5. Can you rewrite the equation in another form before solving it?

Tip: Always double-check calculations when solving for $x$ to avoid errors in larger problems!