https://file.aiquickdraw.com/mathbot/file/1728347051564fr2xwjgb.jpg

Solve and explains the answers to both A and B

Let's solve the problems step by step for both parts A and B based on the image.

### Part (a):

You have a segment \( AC \) with point \( B \) in between. From the image, it seems:

- Segment \( AB = x + 10 \)
- Segment \( BC = 2x + 6 \)
- The whole segment \( AC = 5 \)

Since \( B \) lies on the segment \( AC \), we can use the segment addition postulate:

\[
AB + BC = AC
\]

Substitute the given expressions into this equation:

\[
(x + 10) + (2x + 6) = 5
\]

Now, simplify:

\[
x + 10 + 2x + 6 = 5
\]

\[
3x + 16 = 5
\]

Subtract 16 from both sides:

\[
3x = 5 - 16
\]

\[
3x = -11
\]

Now, divide by 3:

\[
x = \frac{-11}{3}
\]

This is the value of \( x \). To find \( AB \) and \( BC \), substitute \( x = \frac{-11}{3} \) into the expressions for \( AB \) and \( BC \):

\[
AB = \left( \frac{-11}{3} \right) + 10 = \frac{-11}{3} + \frac{30}{3} = \frac{19}{3}
\]

\[
BC = 2\left( \frac{-11}{3} \right) + 6 = \frac{-22}{3} + \frac{18}{3} = \frac{-4}{3}
\]

Thus:
- \( AB = \frac{19}{3} \)
- \( BC = \frac{-4}{3} \)

### Part (b):

Again, you have a segment \( AC \) with point \( B \) in between. The details are:

- Segment \( AB = 9x + 7 \)
- Segment \( BC = -3x + 20 \)
- The whole segment \( AC = 53 \)

By the segment addition postulate:

\[
AB + BC = AC
\]

Substitute the given expressions:

\[
(9x + 7) + (-3x + 20) = 53
\]

Simplify:

\[
9x + 7 - 3x + 20 = 53
\]

\[
6x + 27 = 53
\]

Subtract 27 from both sides:

\[
6x = 53 - 27
\]

\[
6x = 26
\]

Now divide by 6:

\[
x = \frac{26}{6} = \frac{13}{3}
\]

This is the value of \( x \). To find \( AB \) and \( BC \), substitute \( x = \frac{13}{3} \) into the expressions for \( AB \) and \( BC \):

\[
AB = 9\left( \frac{13}{3} \right) + 7 = \frac{117}{3} + 7 = 39 + 7 = 46
\]

\[
BC = -3\left( \frac{13}{3} \right) + 20 = -13 + 20 = 7
\]

Thus:
- \( AB = 46 \)
- \( BC = 7 \)

### Summary of Results:

- (a) \( x = \frac{-11}{3} \), \( AB = \frac{19}{3} \), and \( BC = \frac{-4}{3} \)
- (b) \( x = \frac{13}{3} \), \( AB = 46 \), and \( BC = 7 \)

Would you like further details on any part of this solution, or do you have any questions?

### Related Questions:

1. How do we calculate segment lengths when multiple variables are involved?
2. What does the segment addition postulate imply in geometry?
3. How do we solve equations with fractions and integers combined?
4. Can we always determine the position of points on a line given algebraic expressions?
5. What would happen if the total length was less than the sum of the segments in part (a)?

**Tip:** Always simplify both sides of an equation before isolating the variable to avoid errors.

Solve and explain the answers to both A and B based on the image of two geometric segments with algebraic expressions.

Learn how to solve two linear algebra problems involving geometric segments using the Segment Addition Postulate. This detailed step-by-step solution explains how to find unknown variables in both problems. Suitable for students in Grades 8-10.

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