GMAT Math : DSQ: Calculating the slope of a line
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Example Questions
Example Question #1 : Dsq: Calculating The Slope Of A Line
Is the slope of the line 
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Therefore, the sign of 
The first statement means that 
The second statement - that 
Example Question #2 : Dsq: Calculating The Slope Of A Line
Does a given line with intercepts 
Statement 1:
Statement 2:
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
The slope of a line through 
From Statement 1 alone, we can tell that
so we know the sign of the slope.
From Statement 2 alone, we can tell that
But this can be positive or negative - for example:
but
Example Question #3 : Dsq: Calculating The Slope Of A Line
Does a given line with intercepts
Statement 1:
Statement 2:
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
The slope of a line through
If 
Statement 1 is not enough to determine the sign of 
Case 1:
Case 2:
So if we only know Statement 1, we do not know whether 
If we know both statements, we can solve the system of equations as follows:
Therefore, we know 
Example Question #4 : Dsq: Calculating The Slope Of A Line
Does a given line with intercepts
Statement 1:
Statement 2:
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
The slope of a line through
If
If we know both statements, we try to solve the system of equations as follows:
This means that the system is dependent, and that the statements are essentially the same.
Case 1:
Then
Case 2:
Then
Thus from Statement 1 alone, it cannot be determined whether
Example Question #51 : Lines
You are given two lines. Are they perpendicular?
Statement 1: The sum of their slopes is 
Statement 2: They have the same slope.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
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.
Example Question #6 : Dsq: Calculating The Slope Of A Line
A line includes points 
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
The slope of the line that includes points 
For the question of the sign of the slope to be answered, it must be known whether 
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 
If the two statements together are assumed, we know that 
Example Question #7 : Dsq: Calculating The Slope Of A Line
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: The line is perpendicular to the 
Statement 2: The line has no 
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
The 
A line on the coordinate plane with no 
Example Question #8 : Dsq: Calculating The Slope Of A Line
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: It includes the origin.
Statement 2: It passes through Quadrant II.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
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 
which is a negative slope.
Example Question #9 : Dsq: Calculating The Slope Of A Line
A line is on the coordinate plane. What is its slope?
Statement 1: The line is parallel to the line of the equation 
Statement 2: The line is perpendicular to the line of the equation 
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
If Statement 1 alone holds - that is, if it is known only that the line is parallel to the line of 
If Statement 2 alone holds - that is, if it is known only that the line is perpendicular to the line of the equation 
Either statement alone will yield an answer.
Example Question #10 : Dsq: Calculating The Slope Of A Line
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: The line contains points in both Quadrant I and Quadrant II.
Statement 2: The line contains points in both Quadrant I and Quadrant III.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
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 
which is a positive slope.
Therefore, Statement 2 alone, but not Statement 1 alone, provides a definitive answer.
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