\(\displaystyle Ax + y = C\), in slope-intercept form, is 

\(\displaystyle y = -Ax + C\)

Therefore, the sign of \(\displaystyle -A\) is the sign of the slope.

The first statement means that \(\displaystyle AC\) is positive - all that means is that both \(\displaystyle A\) and \(\displaystyle C\) are nonzero and of like sign. \(\displaystyle A\) can be either positive or negative, and consequently, so can slope \(\displaystyle -A\).

The second statement - that \(\displaystyle A\) is positive - makes \(\displaystyle -A\) , the sign of the slope, negative.

The slope of a line through \(\displaystyle (a,0)\) and \(\displaystyle (0,b)\) is 

\(\displaystyle m = \frac{b-0}{0-a} =- \frac{b}{a}\)

From Statement 1 alone, we can tell that 

\(\displaystyle m =- \frac{b}{a} =- \frac{b}{2b} = - \frac{1}{2} < 0\),

so we know the sign of the slope.

From  Statement 2 alone, we can tell that 

\(\displaystyle m =- \frac{b}{a} =- \frac{b}{b + 4}\)

But this can be positive or negative - for example:

\(\displaystyle b = 4 \Rightarrow m =- \frac{b}{b + 4}=- \frac{4}{4 + 4} =- \frac{4}{8}= - \frac{1}{2} < 0\)

but

\(\displaystyle b = -2 \Rightarrow m =- \frac{b}{b + 4}=- \frac{-2}{-2 + 4} =- \frac{-2}{2}= 1>0\)

The slope of a line through \(\displaystyle (a,0)\) and \(\displaystyle (0,b)\) is 

\(\displaystyle m = \frac{b-0}{0-a} =- \frac{b}{a}\)

If \(\displaystyle a\) and \(\displaystyle b\) have the same sign, then \(\displaystyle m < 0\), making the slope negative; if \(\displaystyle a\) and \(\displaystyle b\) have the same sign, then \(\displaystyle m >0\), making the slope positive.

Statement 1 is not enough to determine the sign of \(\displaystyle m\). 

\(\displaystyle 2a + b = 9\)

\(\displaystyle b = 9 - 2a\)

Case 1: \(\displaystyle a = 1 \Rightarrow b = 9 - 2 \cdot1 = 7\)

Case 2: \(\displaystyle a = 5 \Rightarrow b = 9 - 2 \cdot5 = -1\)

So if we only know Statement 1, we do not know whether \(\displaystyle a\) and \(\displaystyle b\) have the same sign, and, subsequently, we do not know the sign of slope \(\displaystyle m\). A similar argument can be made that Statement 2 provides insufficient information.

If we know both statements, we can solve the system of equations as follows:

\(\displaystyle a - 2b = 17\)

\(\displaystyle a - 2 \left ( 9-2a\right ) = 17\)

\(\displaystyle a -18 +4a= 17\)

\(\displaystyle 5a -18 = 17\)

\(\displaystyle 5a = 35\)

\(\displaystyle a = 7\)

\(\displaystyle b = 9 - 2a = 9 - 2 \cdot 7 = 9 - 14 = -5\)

Therefore, we know \(\displaystyle a\) and \(\displaystyle b\) have unlike sign and \(\displaystyle m >0\).

The slope of a line through \(\displaystyle (a,0)\) and \(\displaystyle (0,b)\) is 

\(\displaystyle m = \frac{b-0}{0-a} =- \frac{b}{a}\)

If \(\displaystyle a\) and \(\displaystyle b\) have the same sign, then \(\displaystyle m < 0\), making the slope negative; if \(\displaystyle a\) and \(\displaystyle b\) have the same sign, then \(\displaystyle m >0\), making the slope positive. 

If we know both statements, we try to solve the system of equations as follows:

\(\displaystyle 3a - 12b = 15\)

\(\displaystyle 3 (4b + 5) - 12b = 15\)

\(\displaystyle 12b+15 - 12b = 15\)

\(\displaystyle 15 = 15\)

This means that the system is dependent, and that the statements are essentially the same.

Case 1: \(\displaystyle b = 1\)

Then \(\displaystyle a = 4b + 5 = 4 \cdot 1 + 5 = 9\)

Case 2: \(\displaystyle b = -1\)

Then \(\displaystyle a = 4b + 5 = 4 \cdot \left (-1 \right ) + 5 = 1\)

Thus from Statement 1 alone, it cannot be determined whether  \(\displaystyle a\) and \(\displaystyle b\) have the same sign, and the sign of the slope cannot be determined. Since Statement 2 is equivalent to Statement 1, the same holds of this statement, as well as both statements together.

Statement 2 alone tells us that the lines are parallel, not perpendicular. Statement 1 alone is neither necessary nor helpful, as the sum of the slopes is irrelevant.

The slope of the line that includes points \(\displaystyle (a,b)\) and \(\displaystyle (c,d)\) is \(\displaystyle m = \frac{b-d}{a -c }\).

For the question of the sign of the slope to be answered, it must be known whether \(\displaystyle a-c\) and \(\displaystyle b-d\) are of the same sign or of different signs, or whether one of them is equal to zero.

Statement 1 alone does not answer this question, as it only states that the denominator is greater; it is possible for this to happen whether both are of like sign or unlike sign. Statement 2 only proves that \(\displaystyle b-d > 0\) - that is, that the denominator is positive.

If the two statements together are assumed, we know that \(\displaystyle a-c > b-d > 0\). Since both the numerator and the denominator are positive, the slope of the line must be positive.

The \(\displaystyle x\)-axis is horizontal, so any line perpendicular to it is vertical and has undefined slope. Statement 1 is sufficient.

A line on the coordinate plane with no \(\displaystyle y\)-intercept does not intersect the \(\displaystyle y\)-axis and therefore must be parallel to it - subsequently, it must be vertical and have undefined slope. This makes Statement 2 sufficient.

Infinitely many lines pass through the origin, and infinitely many lines pass through each quadrant, so neither statement alone is sufficient to answer the question.

Suppose that both statements are known to be true. Since the line passes through quadrant II, it passes through a point \(\displaystyle (-a, b)\) , where \(\displaystyle a,b\) are positive. It also passes through \(\displaystyle (0,0)\) so its slope will be

\(\displaystyle m = \frac{-b-0 }{a-0}= -\frac{b}{a}\)

which is a negative slope.

If Statement 1 alone holds - that is, if it is known only that the line is parallel to the line of \(\displaystyle 5x+ 4y = 20\) - then this equation can be rewritten in slope-intercept form. The slope can be deduced from this, and the two lines, being parallel, will have the same slope.

If Statement 2 alone holds - that is, if it is known only that the line is perpendicular to the line of the equation \(\displaystyle 4x-5y = 40\) - then this equation can be rewritten in slope-intercept form. The slope can be deduced from this, and the first line, which is perpendicular to this one, will have the slope that is the opposite of the reciprocal of that.

Either statement alone will yield an answer.

Examine the diagram below.

It can be seen from the red lines that no conclusions about the sign of the slope of a line can be drawn from Statement 1, since lines of positive, negative, and zero slope can contain points in both Quadrant I and Quadrant II.

If a line contains a point in Quadrant I and a point in Quadrant III, then it contains a point with positive coordinates \(\displaystyle (a,b)\) and a point with negative coordinates \(\displaystyle (-c,-d))\); its slope is

\(\displaystyle m = \frac{b-\left ( -d \right )}{a-\left ( -c \right )}= \frac{b+d}{a+c}\)

which is a positive slope. 

Therefore, Statement 2 alone, but not Statement 1 alone, provides a definitive answer.