 Point S is on line segment start overline, R, T, end overline*RT*. Given S, T, equals, 5, x, minus, 7, comma*ST*=5*x*−7, R, T, equals, 4, x, plus, 9, comma*RT*=4*x*+9, and R, S, equals, 2, x, plus, 7, comma*RS*=2*x*+7, determine the numerical length of start overline, S, T, end overline, .*ST*.

To find the numerical length of \(\overline{ST}\), we can use the given information and apply the segment addition postulate.

### Given:
- \(ST = 5x - 7\)
- \(RT = 4x + 9\)
- \(RS = 2x + 7\)

### Step 1: Apply the segment addition postulate
The segment addition postulate states that the length of segment \(\overline{RT}\) is the sum of the lengths of segments \(\overline{RS}\) and \(\overline{ST}\). Therefore:
\[
RT = RS + ST
\]

### Step 2: Substitute the given values
Substituting the given expressions for \(RT\), \(RS\), and \(ST\):
\[
4x + 9 = (2x + 7) + (5x - 7)
\]

### Step 3: Simplify the equation
First, combine like terms on the right side:
\[
4x + 9 = 2x + 7 + 5x - 7
\]
\[
4x + 9 = 7x
\]

### Step 4: Solve for \(x\)
To isolate \(x\), subtract \(4x\) from both sides:
\[
9 = 3x
\]
Next, divide both sides by 3 to solve for \(x\):
\[
x = 3
\]

### Step 5: Determine the length of \(\overline{ST}\)
Substitute \(x = 3\) into the expression for \(ST\):
\[
ST = 5x - 7 = 5(3) - 7 = 15 - 7 = 8
\]

### Final Answer:
The numerical length of \(\overline{ST}\) is \(\boxed{8}\).

Would you like further details on any step, or do you have any other questions? 

Here are 8 related questions:
1. How do you verify that the value of \(x\) you found satisfies all parts of the problem?
2. What would be the length of \(\overline{RS}\) with the calculated value of \(x\)?
3. How would the equation change if the expressions for \(RS\) or \(ST\) were different?
4. Can you find the length of \(\overline{RT}\) using the value of \(x\) found?
5. Why is it important to combine like terms when solving equations?
6. How does the segment addition postulate help in solving such problems?
7. What would the equation look like if \(\overline{RT}\) were a different expression?
8. How can you solve similar problems involving different geometric figures?

**Tip:** Always check your solutions by substituting the values back into the original expressions to ensure they satisfy all parts of the problem.

Learn how to calculate the length of line segment ST using the segment addition postulate with a step-by-step solution. This problem involves applying geometric principles to determine segment lengths based on given expressions. Suitable for Grades 9-10.

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