M31-30

Official Solution:


If line \(m\) does not pass through the origin, is the slope of line \(m\) negative ?

(1) The product of the x-intercept and the y-intercept of line \(m\) is positive.

Since the product of the intercepts is positive, either both intercepts are negative ((negative, 0) and (0, negative)) or both intercepts are positive ((positive, 0) and (0, positive)). In either case, the line is sloping downwards as we move from left to right, so the slope of line \(m\) is negative. Sufficient.

(2) Line \(m\) passes through the points \((a, \ b)\) and \((c, \ d)\), where \(\frac{b - d}{a-c} < 0 \).

The slope of a line is defined as "rise over run" or the change in \(y\) divided by the change in \(x\) between any two points on the line. In this case, the two points given are \((a,b)\) and \((c,d)\), so the slope of line \(m\) is \(\frac{b - d}{a-c}\), which is given to be negative in this statement. Hence, the slope of line \(m\) is negative. Sufficient.


Answer: D