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In Exercises 9 and 10, point [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex] on [tex]\( \overline{AC} \)[/tex]. Use the information to write an equation in terms of [tex]\( x \)[/tex]. Then solve the equation and find [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex].

9.
[tex]\[
\begin{aligned}
AB &= 13 + 2x \\
BC &= 12 \\
AC &= x + 32
\end{aligned}
\][/tex]

10.
[tex]\[
\begin{aligned}
AB &= 8x + 5 \\
BC &= 5x - 9 \\
AC &= 74
\end{aligned}
\][/tex]



Answer :

GinnyAnswer
Alright, let's address both exercises step-by-step given the information provided.

### Exercise 9

#### Given:
- [tex]\( AB = 13 + 2x \)[/tex]
- [tex]\( BC = 12 \)[/tex]
- [tex]\( AC = x + 32 \)[/tex]

Since point [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex], we can write:
[tex]\[ AB + BC = AC \][/tex]

#### Form the equation:
[tex]\[ (13 + 2x) + 12 = x + 32 \][/tex]

#### Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 13 + 2x + 12 = x + 32 \][/tex]
[tex]\[ 25 + 2x = x + 32 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 25 + x = 32 \][/tex]
Subtract 25 from both sides:
[tex]\[ x = 7 \][/tex]

#### Find [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex]:

- [tex]\( AB = 13 + 2x = 13 + 2(7) = 13 + 14 = 27 \)[/tex]
- [tex]\( BC = 12 \)[/tex] (given)
- [tex]\( AC = x + 32 = 7 + 32 = 39 \)[/tex]

So, the values are:
- [tex]\( x = 7 \)[/tex]
- [tex]\( AB = 27 \)[/tex]
- [tex]\( BC = 12 \)[/tex]
- [tex]\( AC = 39 \)[/tex]

### Exercise 10

#### Given:
- [tex]\( AB = 8x + 5 \)[/tex]
- [tex]\( BC = 5x - 9 \)[/tex]
- [tex]\( AC = 74 \)[/tex]

Since point [tex]\( B \)[/tex] is between [tex]\( A \)[/tex] and [tex]\( C \)[/tex], we can write:
[tex]\[ AB + BC = AC \][/tex]

#### Form the equation:
[tex]\[ (8x + 5) + (5x - 9) = 74 \][/tex]

#### Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 8x + 5 + 5x - 9 = 74 \][/tex]
[tex]\[ 13x - 4 = 74 \][/tex]
Add 4 to both sides:
[tex]\[ 13x = 78 \][/tex]
Divide both sides by 13:
[tex]\[ x = 6 \][/tex]

#### Find [tex]\( AB \)[/tex], [tex]\( BC \)[/tex], and [tex]\( AC \)[/tex]:

- [tex]\( AB = 8x + 5 = 8(6) + 5 = 48 + 5 = 53 \)[/tex]
- [tex]\( BC = 5x - 9 = 5(6) - 9 = 30 - 9 = 21 \)[/tex]
- [tex]\( AC = 74 \)[/tex] (given)

So, the values are:
- [tex]\( x = 6 \)[/tex]
- [tex]\( AB = 53 \)[/tex]
- [tex]\( BC = 21 \)[/tex]
- [tex]\( AC = 74 \)[/tex]

### Summary

For Exercise 9:
- [tex]\( x = 7 \)[/tex]
- [tex]\( AB = 27 \)[/tex]
- [tex]\( BC = 12 \)[/tex]
- [tex]\( AC = 39 \)[/tex]

For Exercise 10:
- [tex]\( x = 6 \)[/tex]
- [tex]\( AB = 53 \)[/tex]
- [tex]\( BC = 21 \)[/tex]
- [tex]\( AC = 74 \)[/tex]

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