https://file.aiquickdraw.com/mathbot/file/1727289080856k2brddkm.jpg

Let's break down the problem step by step.

We are given:

\[
EF = 2x - 12, \quad FG = 3x - 15, \quad \text{and} \quad EG = EF + FG = 23
\]

The total segment \(EG\) is the sum of \(EF\) and \(FG\), so we can set up the following equation:

\[
(2x - 12) + (3x - 15) = 23
\]

Now, simplify the left side:

\[
2x - 12 + 3x - 15 = 23
\]

Combine like terms:

\[
5x - 27 = 23
\]

Add 27 to both sides:

\[
5x = 50
\]

Now, divide by 5:

\[
x = 10
\]

Now that we know \(x = 10\), substitute this value into the expressions for \(EF\) and \(FG\):

\[
EF = 2(10) - 12 = 20 - 12 = 8
\]
\[
FG = 3(10) - 15 = 30 - 15 = 15
\]

Thus, the values are:
- \(x = 10\)
- \(EF = 8\)
- \(FG = 15\)

The correct answer is:
\[
x = 10, \quad EF = 8, \quad FG = 15
\]

This corresponds to the first option: \(x = 10, EF = 8, FG = 15\).

Would you like further details or clarification? Here are five related questions to expand this concept:

1. How do you calculate the length of a segment when multiple subsegments are given algebraically?
2. What happens if the sum of \(EF + FG\) is less than the total \(EG\)?
3. How would the solution change if the equations for \(EF\) and \(FG\) involved different coefficients for \(x\)?
4. How can you visualize this problem on a number line or with a geometric drawing?
5. What strategies can be used to check if the solution to the system of equations is correct?

**Tip:** Always ensure that the total segment length matches the sum of the individual parts when solving segment problems in geometry!

If EF = 2x - 12, FG = 3x - 15, and EG = 23, find the values of x, EF, and FG.

Solve the algebraic problem where EF = 2x - 12, FG = 3x - 15, and EG = 23. This problem illustrates how to apply linear equations to find the lengths of segments EF and FG on a line. Suitable for students in grades 8-10.

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