The graph of the line ax + by = c is drawn, where a, b and c are real non-zero constants.
Which one of the following is a necessary but not sufficient condition for the line to have a positive gradient and a positive y-intercept?
- Acb > 0 and ab < 0
- Bcb < 0 and ab > 0
- Ca > b > c
- Da < b < c
- Ea and c have opposite signs
- Fa and c have the same sign
Show the answer and worked solution
answer · E
- Acb > 0 and ab < 0
- Bcb < 0 and ab > 0
- Ca > b > c
- Da < b < c
- Ea and c have opposite signs
- Fa and c have the same sign
Rearranged, the line is y = −abx + cb, so a positive gradient means ab < 0 and a positive intercept means cb > 0. Together those say a and b have opposite signs while c and b share a sign — so a and c have opposite signs, making that condition necessary. It is not sufficient: a = 1, b = 1, c = −1 has a and c opposite in sign but gives gradient −1. (The first option is the full condition, so it is sufficient too.)