# GMAT Math : DSQ: Calculating the slope of a line

## Example Questions

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### Example Question #661 : Geometry

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 sufficient to answer the question, but NEITHER 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.

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.

Explanation:

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 #51 : Lines

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  in the triangle.

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.

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.

Explanation:

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  - and it depends on the number in the square, , and the triangle . Each statement alone gives the number in only one of the shapes; the two statements together give the numbers in both shapes and allow the slope to be calculated, thereby answering the question of Quinn's success.

### Example Question #781 : Data Sufficiency Questions

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.

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.

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.

Explanation:

Replacing the shapes with variables, the template becomes

Divide by  to get the equation in slope-intercept form:

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, , and . Ava succeeded.

### Example Question #61 : Lines

John's teacher gave him two equations, each with one coefficient missing, as follows:

John was challenged to write one number in each shape in order to form two equations whose lines have the same slope. Did he succeed?

Statement 1: The number John wrote in the box is three times the number he wrote in the circle.

Statement 2: John wrote  in the circle.

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.

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.

Explanation:

Rewrite the two equations with variables rather than shapes:.

The first equation can be rewritten in slope-intercept form:

Its line has slope is , so if the number in the square is known, the slope is known.

is already in slope-intercept form; its line has slope , the number in the circle.

Statement 2 alone gives the number in the circle but provides no clue to the number in the square.

Now assume Statement 1 alone. Then . The slope of the first line is

,

the slope of the second line. Statement 1 provides sufficient proof that John was successful.

### Example Question #661 : Geometry

Does a line on the coordinate plane have undefined slope?

Statement 1: It has -intercept

Statement 2: It passes through Quadrant II.

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.

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.

Explanation:

A line with undefined slope is a vertical line.

Infinitely many lines, some vertical and some not, pass through , and infinitely many lines, some vertical and some not, pass through each quadrant, so neither statement alone is sufficient to answer the question.

Now assume both statements are true. Then, since the line passes through Quadrant II, it passes through at least one point with negative -coordinate and positive -coordinate, which we call . Its slope will be

,

a negative slope. Therefore, the slope is not undefined.

### Example Question #781 : Data Sufficiency Questions

Tim 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 Tim succeed?

Statement 1: Tim wrote  in the square.

Statement 2: Tim wrote  in the circle.

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.

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 insufficient to answer the question.

Explanation:

Assume both statements are true. The equation takes the form

,

where  is the number Tim wrote in the triangle.

Put the equation in slope-intercept form:

The coefficient of , which is , is the slope. As can be seen, the value of —that is, the number Tim wrote in the triangle—needs to be known. However, this information is still not given. Whether Tim succeeded is unknown.

### Example Question #662 : Geometry

Marisol was challenged by her teacher to fill in the square and the circle in the diagram below with two numbers to form the equation of a line with slope .

Did Marisol succeed?

Statement 1: Marisol wrote a  in the box.

Statement 2: Marisol wrote a  in the circle.

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.

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.

Explanation:

The pattern shows a linear equation in slope-intercept form, , with slope  represented by the square and -intercept  represented by the circle. Statement 1 gives the slope, so it provides sufficient information; Statement 2 gives only the -intercept, so it is unhelpful.

### Example Question #671 : Geometry

Veronica's teacher gave her two equations, the first with the coefficients of both variables missing, as follows:

Veronica was challenged to write one number in each shape in order to form an equation whose line has the same slope as that of the second equation. The only restriction was that she could not write a 5 in the square or a 3 in the circle.

Did Veronica write an equation with the correct slope?

Statement 1: Veronica wrote a negative integer in the square and a positive integer in the circle.

Statement 2: Veronica wrote an 8 in the circle.

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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The slope of the line of

can be found by writing this equation in slope-intercept form:

The slope of the line is the coefficient of  is , so Veronica must place the numbers in the shapes to yield an equation whose slope has this equation.

Rewrite the top equation as

The slope, in terms of  and , can be found similarly:

Its slope is .

Statement 1 asserts that  and  are of unlike sign, so the slope  must be negative. It cannot have sign , so the question is answered.

Assume Statement 2 alone. Then in the above equation, , so the slope is . The slope now depends on the value of , so Statement 2 gives insufficient information.

### Example Question #672 : Geometry

Give the slope of a line on the coordinate plane.

Statement 1: The line passes through the graph of the equation  on the -axis.

Statement 2: The line passes through the graph of the equation  on the -axis.

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.

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.

Explanation:

The -intercept(s) of the graph of  can be found by setting  and solving for :

The graph has exactly one -intercept, .

The -intercept(s) of the graph of  can be found by setting  and solving for :

The graph has exactly one -intercept, .

In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. Each of the two statements yields one of the points, so neither statement alone is sufficient to determine the slope. The two statements together, however, yield two points, and are therefore enough to determine the slope.

### Example Question #61 : Lines

Give the slope of a line on the coordinate plane.

Statement 1: The line passes through the vertex of the parabola of the equation .

Statement 2: The line passes through the -intercept of the parabola of the equation .

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 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.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The vertex of the parabola of the equation  can be found by first taking , then substituting in the equation and solving for .

The vertex is the point . Since , this is also the -intercept.

In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. From the two statements together, we only know the -intercept and the vertex; however, they are one and the same. Therefore, we have insufficient information to find the slope.

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