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The final answer, following the maze correctly, is [tex]\( x = 31 \)[/tex]. All other equations result in [tex]\( x = 12 \)[/tex].
1. Start: [tex]\( x + 2 = 12 \)[/tex]
- Subtract 2 from both sides to isolate x:
[tex]- \( x = 12 - 2 \)[/tex]
[tex]- \( x = 10 \)[/tex]
2. [tex]\( x + 7 = 17 \)[/tex]
- Subtract 7 from both sides:
[tex]- \( x = 17 - 7 \)[/tex]
[tex]- \( x = 10 \)[/tex]
3. [tex]\( x - 3 = 9 \)[/tex]
- Add 3 to both sides:
[tex]- \( x = 9 + 3 \)[/tex]
[tex]- \( x = 12 \)[/tex]
4. [tex]\( x + 3 = 15 \)[/tex]
- Subtract 3 from both sides:
[tex]- \( x = 15 - 3 \)[/tex]
[tex]- \( x = 12 \)[/tex]
5. [tex]\( x - 1 = 11 \)[/tex]
- Add 1 to both sides:
[tex]- \( x = 11 + 1 \)[/tex]
[tex]- \( x = 12 \)[/tex]
6. [tex]\( x + 12 = 24 \)[/tex]
- Subtract 12 from both sides:
[tex]- \( x = 24 - 12 \)[/tex]
[tex]- \( x = 12 \)[/tex]
7. [tex]\( x - 2 = 14 \)[/tex]
- Add 2 to both sides:
[tex]- \( x = 14 + 2 \)[/tex]
[tex]- \( x = 16 \)[/tex] (This seems to be a mistake as it does not match the pattern of the other solutions. Please double-check this equation.)
8. [tex]\( x + 5 = 19 \)[/tex]
- Subtract 5 from both sides:
[tex]- \( x = 19 - 5 \)[/tex]
[tex]- \( x = 14 \)[/tex] (This also seems incorrect based on the pattern.)
9. [tex]\( x - 6 = 12 \)[/tex]
- Add 6 to both sides:
[tex]- \( x = 12 + 6 \)[/tex]
[tex]- \( x = 18 \)[/tex] (Another discrepancy here.)
10. [tex]\( x + 4 = 16 \)[/tex]
- Subtract 4 from both sides:
[tex]- \( x = 16 - 4 \)[/tex]
[tex]- \( x = 12 \)[/tex]
11. [tex]\( x + 8 = 32 \)[/tex]
- Subtract 8 from both sides:
[tex]- \( x = 32 - 8 \)[/tex]
[tex]- \( x = 24 \)[/tex] (This does not seem correct.)
12.[tex]\( x + 6 = 30 \)[/tex]
- Subtract 6 from both sides:
[tex]- \( x = 30 - 6 \)[/tex]
[tex]- \( x = 24 \)[/tex] (This also seems incorrect.)
Finish: [tex]\( x + 31 = 62 \)[/tex]
- Subtract 31 from both sides:
[tex]- \( x = 62 - 31 \)[/tex]
[tex]- \( x = 31 \)[/tex] (This is the final answer if we follow the maze correctly.)
It appears there may be some inconsistencies in the equations provided. The pattern suggests that the solution to each equation should be [tex]\( x = 12 \),[/tex] except for the last one which is [tex]\( x = 31 \)[/tex].
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Rewritten by : Barada
To solve equations in problems involving parallel lines cut by a transversal, one should identify the angle relationships (corresponding, alternate interior, alternate exterior, consecutive interior), set up equations based on these relationships in a chosen coordinate system, and solve these equations progressively. The solutions should be verified by substituting them back into the original equations.
In the context of Parallel Lines cut by a Transversal in geometry, we generally look for angle relationships and use them to solve for unknown values. The key here is to identify corresponding angles, alternate interior angles, alternate exterior angles, or consecutive interior angles, which have specific relationships when lines are cut by a transversal.
For example, corresponding angles are equal, alternate interior and alternate exterior angles are equal, whereas consecutive interior angles are supplementary (add up to 180 degrees).
The task here is first, to pick a convenient coordinate system (typically one horizontal axis - x, and one vertical axis - y), then identify the relationships between the angles formed by the parallel lines and the transversal, set up equations based on these relationships, and solve them. As you proceed, each solved equation should simplify the subsequent ones.
Remember to verify your solutions by inserting them back into your original equations to confirm they hold true.
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