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We are given three lengths:
- \( ST = 9x \)
- \( TU = 5x + 6 \)
- \( SU = 12x + 16 \)

Since \( S, T, U \) are points on a line, the sum of the lengths \( ST \) and \( TU \) must equal the total length \( SU \). So, we can set up the equation:

\[
ST + TU = SU
\]

Substituting the given values:

\[
9x + (5x + 6) = 12x + 16
\]

Let's solve this equation step by step.

1. Combine like terms on the left-hand side:

\[
9x + 5x + 6 = 12x + 16
\]

\[
14x + 6 = 12x + 16
\]

2. Subtract \( 12x \) from both sides:

\[
14x - 12x + 6 = 16
\]

\[
2x + 6 = 16
\]

3. Subtract 6 from both sides:

\[
2x = 10
\]

4. Divide by 2:

\[
x = 5
\]

Now that we know \( x = 5 \), we can substitute this value back into the expression for \( ST \):

\[
ST = 9x = 9 \times 5 = 45
\]

So, \( ST = 45 \).

Would you like more details or have any other questions? Here are 5 related problems:

1. If \( SU = 12x + 16 \), what is the length of \( SU \) when \( x = 5 \)?
2. What is the length of \( TU \) when \( x = 5 \)?
3. How would the equation change if \( SU = 15x + 20 \) instead of \( 12x + 16 \)?
4. If the total length of \( SU \) is fixed, how does changing \( x \) affect the relative lengths of \( ST \) and \( TU \)?
5. Can we determine the exact position of points \( S, T, U \) on a number line if given additional coordinates?

**Tip:** Always double-check the algebraic steps, especially when combining like terms and solving for variables.

If ST = 9x, TU = 5x + 6, and SU = 12x + 16, what is ST?

Learn how to solve the equation where ST = 9x, TU = 5x + 6, and SU = 12x + 16. Understand how to use algebraic methods to find the value of ST by solving for x in a system of linear equations. Suitable for Grades 8-10.

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