To determine which two lines are parallel, we need to identify pairs of lines that have the same slope.
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Line equations given:
- \( y = -12x + 6 \) (slope = -12)
- \( y = 12x - 8 \) (slope = 12)
- \( y = -\frac{1}{2}x + 6 \) (slope = -\frac{1}{2})
- \( y = \frac{1}{2}x - 8 \) (slope = \frac{1}{2})
- \( y = 13x + 4 \) (slope = 13)
- \( y = 13x + 4 \) (slope = 13)
- \( y = \frac{1}{3}x + 4 \) (slope = \frac{1}{3})
- \( y = \frac{1}{3}x + 4 \) (slope = \frac{1}{3})
- \( y = 23x - 6 \) (slope = 23)
- \( y = -\frac{3}{2}x - 6 \) (slope = -\frac{3}{2})
- \( y = \frac{2}{3}x - 6 \) (slope = \frac{2}{3})
- \( y = -\frac{3}{2}x - 6 \) (slope = -\frac{3}{2})
- \( y = 12x + 3 \) (slope = 12)
- \( y = 12x - 5 \) (slope = 12)
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Identifying parallel lines:
- The equations \( y = 13x + 4 \) and \( y = 13x + 4 \) are clearly parallel because they have the same slope (13).
- The equations \( y = \frac{1}{3}x + 4 \) and \( y = \frac{1}{3}x + 4 \) are also parallel because they have the same slope (\(\frac{1}{3}\)).
- The equations \( y = 12x + 3 \) and \( y = 12x - 5 \) are parallel because they have the same slope (12).
In conclusion, the two lines that are parallel from the above options are:
- \( y = 12x + 3 \) and \( y = 12x - 5 \).
Alternatively, we may also identify \( y = 13x + 4 \) and \( y = 13x + 4 \) as both lines are identical and therefore parallel.
So the two sets of parallel lines are:
- \( y = 12x + 3 \) and \( y = 12x - 5 \)
- \( y = 13x + 4 \) and \( y = 13x + 4 \) (identical, hence parallel)
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