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Points A, B, and C are collinear. Point B is between A and C. Solve for [tex]\(x\)[/tex].

7. [tex]\(AC = -3 + 6x\)[/tex], [tex]\(AB = 3x - 3\)[/tex], and [tex]\(BC = 9\)[/tex]. Find [tex]\(x\)[/tex].

8. [tex]\(AB = 2x + 22\)[/tex], [tex]\(BC = 3\)[/tex], and [tex]\(AC = x + 17\)[/tex]. Find [tex]\(x\)[/tex].

9. Find [tex]\(x\)[/tex] if [tex]\(AB = x + 12\)[/tex], [tex]\(BC = x + 12\)[/tex], and [tex]\(AC = 14\)[/tex].

10. Find [tex]\(x\)[/tex] if [tex]\(AC = 22\)[/tex], [tex]\(AB = x + 22\)[/tex], and [tex]\(BC = x + 20\)[/tex].

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Points A, B, and C are collinear. Point B is between A and C. Find the length indicated.

11. [tex]\(AC = 3x - 5\)[/tex], [tex]\(BC = 10\)[/tex], and [tex]\(AB = -7 + 2x\)[/tex]. Find [tex]\(AB\)[/tex].

12. [tex]\(AB = 9\)[/tex], [tex]\(AC = 5x - 1\)[/tex], and [tex]\(BC = 2x + 2\)[/tex]. Find [tex]\(BC\)[/tex].



Answer :

GinnyAnswer
Sure! Let's solve each question step-by-step.

### 7. Given:
- [tex]\( AC = -3 + 6x \)[/tex]
- [tex]\( AB = 3x - 3 \)[/tex]
- [tex]\( BC = 9 \)[/tex]

Since A, B, and C are collinear and B is between A and C, we have:
[tex]\[ AB + BC = AC \][/tex]
Substituting the given values:
[tex]\[ (3x - 3) + 9 = -3 + 6x \][/tex]

Solving this equation for [tex]\( x \)[/tex]:

[tex]\[ 3x - 3 + 9 = -3 + 6x \][/tex]
[tex]\[ 3x + 6 = -3 + 6x \][/tex]
[tex]\[ 6 + 3 = 3x \][/tex]
[tex]\[ 9 = 3x \][/tex]
[tex]\[ x = 3 \][/tex]

### 8. Given:
- [tex]\( AB = 2x + 22 \)[/tex]
- [tex]\( BC = 3 \)[/tex]
- [tex]\( AC = x + 17 \)[/tex]

Again, since A, B, and C are collinear and B is between A and C:
[tex]\[ AB + BC = AC \][/tex]
Substituting the given values:
[tex]\[ (2x + 22) + 3 = x + 17 \][/tex]
[tex]\[ 2x + 25 = x + 17 \][/tex]
[tex]\[ 2x - x = 17 - 25 \][/tex]
[tex]\[ x = -8 \][/tex]

### 9. Given:
- [tex]\( AB = x + 12 \)[/tex]
- [tex]\( BC = x + 12 \)[/tex]
- [tex]\( AC = 14 \)[/tex]

Since A, B, and C are collinear and B is between A and C:
[tex]\[ AB + BC = AC \][/tex]
Substituting the given values:
[tex]\[ (x + 12) + (x + 12) = 14 \][/tex]
[tex]\[ 2x + 24 = 14 \][/tex]
[tex]\[ 2x = 14 - 24 \][/tex]
[tex]\[ 2x = -10 \][/tex]
[tex]\[ x = -5 \][/tex]

### 10. Given:
- [tex]\( AC = 22 \)[/tex]
- [tex]\( AB = x + 22 \)[/tex]
- [tex]\( BC = x + 20 \)[/tex]

Since A, B, and C are collinear and B is between A and C:
[tex]\[ AB + BC = AC \][/tex]
Substituting the given values:
[tex]\[ (x + 22) + (x + 20) = 22 \][/tex]
[tex]\[ 2x + 42 = 22 \][/tex]
[tex]\[ 2x = 22 - 42 \][/tex]
[tex]\[ 2x = -20 \][/tex]
[tex]\[ x = -10 \][/tex]

### 11. Given:
- [tex]\( AC = 3x - 5 \)[/tex]
- [tex]\( BC = 10 \)[/tex]
- [tex]\( AB = -7 + 2x \)[/tex]

We are asked to find [tex]\( \dot{A B} \)[/tex], which means solving for [tex]\( x \)[/tex] first:
Since A, B, and C are collinear and B is between A and C:
[tex]\[ AB + BC = AC \][/tex]
Substituting the given values:
[tex]\[ (-7 + 2x) + 10 = 3x - 5 \][/tex]
[tex]\[ 2x + 3 = 3x - 5 \][/tex]
[tex]\[ 3 + 5 = 3x - 2x \][/tex]
[tex]\[ 8 = x \][/tex]
Thus,
[tex]\[ A \dot{B} = 8 \][/tex]

### 12. Given:
- [tex]\( AB = 9 \)[/tex]
- [tex]\( AC = 5x - 1 \)[/tex]
- [tex]\( BC = 2x + 2 \)[/tex]

We are asked to find [tex]\( \dot{B C} \)[/tex], which means solving for [tex]\( x \)[/tex] first:
Since A, B, and C are collinear and B is between A and C:
[tex]\[ AB + BC = AC \][/tex]
Substituting the given values:
[tex]\[ 9 + (2x + 2) = 5x - 1 \][/tex]
[tex]\[ 11 + 2x = 5x - 1 \][/tex]
[tex]\[ 11 + 1 = 5x - 2x \][/tex]
[tex]\[ 12 = 3x \][/tex]
[tex]\[ x = 4 \][/tex]
Thus,
[tex]\[ B \dot{C} = 4 \][/tex]

So, the solutions for each part are:
[tex]\[ \text{7) } x = 3 \][/tex]
[tex]\[ \text{8) } x = -8 \][/tex]
[tex]\[ \text{9) } x = -5 \][/tex]
[tex]\[ \text{10) } x = -10 \][/tex]
[tex]\[ \text{11) } x = 8 \][/tex]
[tex]\[ \text{12) } x = 4 \][/tex]

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