To determine which two lines are parallel, we need to identify pairs of lines that have the same slope. Lines are parallel if they have the same slope but different y-intercepts.
Let's analyze the given equations:
- \(y = 23x - 6\) (slope = 23) and \(y = -32x - 6\) (slope = -32)
- \(y = \frac{2}{3}x - 6\) (slope = \(\frac{2}{3}\)) and \(y = -\frac{3}{2}x - 6\) (slope = -\(\frac{3}{2}\))
- \(y = 12x + 3\) (slope = 12) and \(y = 12x - 5\) (slope = 12)
- \(y = \frac{1}{2}x + 3\) (slope = \(\frac{1}{2}\)) and \(y = \frac{1}{2}x - 5\) (slope = \(\frac{1}{2}\))
- \(y = -12x + 6\) (slope = -12) and \(y = 12x - 8\) (slope = 12)
- \(y = -\frac{1}{2}x + 6\) (slope = -\(\frac{1}{2}\)) and \(y = \frac{1}{2}x - 8\) (slope = \(\frac{1}{2}\))
- \(y = 13x + 4\) and \(y = 13x + 4\) (slope = 13) - these lines are identical.
The lines that are parallel must have the same slope:
- From the analysis, the pairs of lines with the same slopes are:
- \(y = 12x + 3\) and \(y = 12x - 5\) (both with slope = 12)
- \(y = \frac{1}{2}x + 3\) and \(y = \frac{1}{2}x - 5\) (both with slope = \(\frac{1}{2}\))
- The lines \(y = 13x + 4\) and \(y = 13x + 4\) are also parallel since they are the same line.
Thus, any of the pairs involving the same slopes can be identified as parallel.
The final answers are:
- \(y = 12x + 3\) and \(y = 12x - 5\) (parallel)
- \(y = \frac{1}{2}x + 3\) and \(y = \frac{1}{2}x - 5\) (parallel)
- \(y = 13x + 4\) and \(y = 13x + 4\) (the same line, hence parallel)
Choose any of these pairs according to your requirement. A clear pair would be:
y = 12x + 3 and y = 12x - 5 are one set of parallel lines.