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The two functions in the question only take values between 5 and 155, so plotchooses a corresponding y range.

There is a space between the x axis and the y axis to indicate that the y range does not start at zero. In the example from the question, it turns out that the space between the x axis and the y axis is larger than the missing part of the y axis...

I assume this is what you find puzzling, and "not to break the y-axis" means the y range should include 0 in this case.

This can be achieved here by specifying ymin=0, either in the plot:

sage: plot([12 * x + 23, 15 * x + 5], (x, 0, 10), aspect_ratio=.05, ymin=0)

or in the show:

sage: p = plot([12 * x + 23, 15 * x + 5], (x, 0, 10), aspect_ratio=.05)
sage: p.show(ymin=0)

If the graph was below 0, one would instead use ymax=0.

To not depend on whether the plot is above or below 0, add a transparent point at the origin to the plot.

sage: p = plot([12 * x + 23, 15 * x + 5], (x, 0, 10), aspect_ratio=.05)
sage: origin = point2d([(0, 0)], alpha=0)
sage: (origin + p).show()