GMAT Math : Lines
Study concepts, example questions & explanations for GMAT Math
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Example Questions
Example Question #1 : Dsq: Calculating The Slope Of A Line
Does a given line with intercepts 
Statement 1:
Statement 2:
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.
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.
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 #771 : Data Sufficiency Questions
You are given two lines. Are they perpendicular?
Statement 1: The sum of their slopes is 
Statement 2: They have the same slope.
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.
EITHER 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.
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:
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.
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 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 #652 : Geometry
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 
Statement 1 ALONE is sufficient to answer the question, but Statement 2 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.
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 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 #661 : Geometry
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.
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.
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 #662 : Geometry
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 
Statement 1 ALONE is sufficient to answer the question, but Statement 2 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.
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.
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 #663 : Geometry
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.
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.
BOTH statements TOGETHER are insufficient 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.
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.
Example Question #21 : Other Lines
Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?
Statement 1: It passes through the point 
Statement 2: It passes through Quadrant III.
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 insufficient to answer the question.
Examine the diagram below.
Both the red line and the green line fit the descriptions in both statements. The red line has negative slope and the green line has positive slope.
The two statements together give insufficient infomation.
Example Question #665 : Geometry
Quinn was challenged by his teacher to fill in the square, the triangle, and the circle in the diagram below with three numbers to form the equation of a line with slope 
Did Quinn succeed?
Statement 1: Quinn wrote a 3 in the square.
Statement 2: Quinn wrote a 
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.
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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER 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 equation is in standard form, with shapes replacing coefficients; that is, the equation is
Rewrite the equation in slope-intercept form:
The slope of the line of the equation is 
Example Question #142 : Coordinate Geometry
Ava was challenged by her teacher to fill in the triangle, the square, and the circle in the diagram below with three numbers to form the equation of a line with slope 1.
Did Ava succeed?
Statement 1: Ava wrote a 1 in the circle.
Statement 2: Ava wrote the same positive number in both the triangle and the square.
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.
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.
Replacing the shapes with variables, the template becomes
Divide by 
The slope is 
The slope is the ratio of the number in the square to the number in the triangle, so the number in the circle is irrelevant, making Statement 1 unhelpful.
Assume Statement 2 alone. Since the numbers in the square and the triangle are equal, 
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