The problem from the image reads:

Given $S$ is between $T$ and $U$, $TS = SU$, $TU = 5x - 4$, and $SU = x + 7$. Find $TS$.

Let's solve this step by step.

Step 1: Express $TU$ in terms of $TS$ and $SU$

Since $S$ is between $T$ and $U$, and $TS = SU$, we know: $TU = TS + SU$ Given $TS = SU$, we can substitute: $TU = TS + TS = 2TS$ Therefore: $TU = 2TS$

Step 2: Use the given expressions for $TU$ and $SU$

We're given: $TU = 5x - 4$ and $SU = x + 7$ Since $TS = SU$, we can substitute $TS = x + 7$ into the equation $TU = 2TS$: $5x - 4 = 2(x + 7)$

Step 3: Solve for $x$

Expanding the equation: $5x - 4 = 2x + 14$ Now, subtract $2x$ from both sides: $3x - 4 = 14$ Add 4 to both sides: $3x = 18$ Finally, divide by 3: $x = 6$

Step 4: Find $TS$

Since $TS = x + 7$, substitute $x = 6$: $TS = 6 + 7 = 13$

Final Answer

The length of $TS$ is 13.

Would you like more details or have any questions? Here are some related questions:

1. How would you find $TU$ after determining $x$?
2. What would happen if $TS$ was not equal to $SU$?
3. Can this method be applied to other similar geometric problems?
4. How would you verify the solution by substituting back into the original equation?
5. What if $S$ was not between $T$ and $U$, how would that change the problem?

Tip: When working with geometric problems involving segments, always double-check the relationships between the points and the given information before solving equations.