Point B is on line segment start overline, A, C, end overline
AC
. Given A, C, equals, 2, x, plus, 7, commaAC=2x+7, B, C, equals, x, commaBC=x, and A, B, equals, 5, x, minus, 9, commaAB=5x−9, determine the numerical length of start overline, A, B, end overline, .
AB
.

Given the problem, we have a line segment \( AC \) with a point \( B \) on it. The lengths of the segments are provided as:

- \( AC = 2x + 7 \)
- \( BC = x \)
- \( AB = 5x - 9 \)

Since \( B \) is on the line segment \( AC \), the total length \( AC \) should be the sum of the lengths of \( AB \) and \( BC \). Therefore, we can write the equation:

\[
AB + BC = AC
\]

Substituting the given expressions:

\[
(5x - 9) + x = 2x + 7
\]

Now, let's solve this equation step by step:

1. Combine like terms on the left side:

\[
5x + x - 9 = 2x + 7
\]

\[
6x - 9 = 2x + 7
\]

2. Move all terms involving \( x \) to one side by subtracting \( 2x \) from both sides:

\[
6x - 2x - 9 = 7
\]

\[
4x - 9 = 7
\]

3. Add 9 to both sides to isolate the term with \( x \):

\[
4x = 16
\]

4. Divide both sides by 4:

\[
x = 4
\]

Now that we have \( x = 4 \), we can find the numerical length of \( AB \) by substituting \( x \) back into the expression for \( AB \):

\[
AB = 5x - 9 = 5(4) - 9 = 20 - 9 = 11
\]

Thus, the numerical length of \( AB \) is \( \boxed{11} \).

Would you like further details or have any questions?

Here are 5 questions related to this problem:

1. What would the length of \( BC \) be if \( x = 4 \)?
2. How does the relationship \( AB + BC = AC \) help solve for \( x \)?
3. What happens if \( B \) were not on the line segment \( AC \)?
4. How would the solution change if \( AB = 5x + 9 \) instead?
5. Can you verify the solution by calculating the length of \( AC \) using \( x = 4 \)?

**Tip:** Always check your final answer by substituting the value back into the original equation to ensure consistency.

Learn how to calculate the length of line segment AB on line segment AC using algebraic equations. Understand the relationship between segments and solve for the unknown variable x. Suitable for students in Grades 7-9.

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