GMAT Math : Other Lines

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Example Question #11 : 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 passes through the point (-7,0) .

Statement 2: It passes through Quadrant III.

Possible Answers:

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Examine the diagram below.

Axes_3

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.

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Example Question #12 : Dsq: Calculating The Slope Of A Line

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 $\frac{3}{5}$ .

$\square x + \bigtriangleup y = \bigcirc$

Did Quinn succeed?

Statement 1: Quinn wrote a 3 in the square.

Statement 2: Quinn wrote a in the triangle.

Possible Answers:

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.

Correct answer:

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

Ax+By = C .

Rewrite the equation in slope-intercept form:

By = -Ax + C

$\frac{By }{B}=\frac{ -Ax + C}{B}$

$y =-\frac{ A}{B}x + \frac{C}{B}$

The slope of the line of the equation is $-\frac{A}{B}$ - 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.

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Example Question #13 : Dsq: Calculating The Slope Of A Line

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.

$\bigtriangleup y = \square x + \bigcirc$

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.

Possible Answers:

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.

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.

Correct answer:

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

My = Nx +P

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

$\frac{My}{M} = \frac{Nx +P}{M}$

$y = \frac{N }{M}x+\frac{P}{M}$

The slope is $\frac{N }{M}$ .

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, M = N , and $\frac{N }{M}= 1$ . Ava succeeded.

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Example Question #21 : Other Lines

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

$\square x - 3 y = 9$

$y = \bigcirc x - 7$

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 $\frac{1}{3}$ in the circle.

Possible Answers:

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.

Correct answer:

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:

Ax -3 y = 9

-3 y = -Ax +9

$\frac{-3 y}{-3} =\frac{ -Ax +9}{-3}$

$y =\frac{ A }{3}x-3$

Its line has slope is $\frac{ A }{3}$ , so if the number in the square is known, the slope is known.

y = B x - 7

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 A = 3B . The slope of the first line is

$\frac{ A }{3} = \frac{ 3B }{3} = B$ ,

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

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Example Question #15 : Dsq: Calculating The Slope Of A Line

Does a line on the coordinate plane have undefined slope?

Statement 1: It has -intercept (5,0)

Statement 2: It passes through Quadrant II.

Possible Answers:

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.

Correct answer:

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 (5,0) , 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 (-a, b) . Its slope will be

$m = \frac{0 - b }{5 - (-a)} = \frac{ - b }{5 +a} = -\frac{ b }{5 +a}$ ,

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

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Example Question #16 : Dsq: Calculating The Slope Of A Line

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 $\frac{2}{3}$ .

$\square x + \bigtriangleup y = \bigcirc$

Did Tim succeed?

Statement 1: Tim wrote $\frac{2}{3}$ in the square.

Statement 2: Tim wrote $\frac{2}{3}$ in the circle.

Possible Answers:

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are true. The equation takes the form

$\frac{2}{3} x +Ay = \frac{2}{3}$ ,

where is the number Tim wrote in the triangle.

Put the equation in slope-intercept form:

$Ay =- \frac{2}{3} x + \frac{2}{3}$

$y =- \frac{2}{3A} x + \frac{2}{3A}$

The coefficient of , which is $- \frac{2}{3A}$ , 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.

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Example Question #17 : Dsq: Calculating The Slope Of A Line

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 .

$y = \square x + \bigcirc$

Did Marisol succeed?

Statement 1: Marisol wrote a in the box.

Statement 2: Marisol wrote a in the circle.

Possible Answers:

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.

Correct answer:

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, y = mx+b , 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.

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Example Question #18 : Dsq: Calculating The Slope Of A Line

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

$\square x- \bigcirc y = 9$

5 x - 3 y = 9

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.

Possible Answers:

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.

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.

Correct answer:

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

5 x - 3 y = 9

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

- 3 y =-5x+ 9

$\frac{- 3 y}{-3} =\frac{-5x+ 9}{-3}$

$y=\frac{-5x }{-3} + \frac{ 9}{-3}$

$y=\frac{5 }{3} x-3$

The slope of the line is the coefficient of is $\frac{5}{3}$ , so Veronica must place the numbers in the shapes to yield an equation whose slope has this equation.

Rewrite the top equation as

A x-By = 9

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

-By =- A x+ 9

$\frac{-By }{-B}=\frac{- A x+ 9 }{-B}$

$y=\frac{- A x }{-B} + \frac{9}{-B}$

$y=\frac{ A }{ B} x- \frac{9}{B}$

Its slope is $\frac{A}{B}$ .

Statement 1 asserts that and are of unlike sign, so the slope $\frac{A}{B}$ must be negative. It cannot have sign $\frac{5}{3}$ , so the question is answered.

Assume Statement 2 alone. Then in the above equation, B=8 , so the slope is $\frac{A}{8}$ . The slope now depends on the value of , so Statement 2 gives insufficient information.

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Example Question #61 : Lines

Give the slope of a line on the coordinate plane.

Statement 1: The line passes through the graph of the equation y = |x+1| on the -axis.

Statement 2: The line passes through the graph of the equation y = |x-1| on the -axis.

Possible Answers:

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.

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.

Correct answer:

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 y = |x+1| can be found by setting y = 0 and solving for :

y = |x+1|

0= |x+1|

x+1 = 0

x = -1

The graph has exactly one -intercept, (-1,0) .

The -intercept(s) of the graph of y = |x-1| can be found by setting x= 0 and solving for :

y = |x-1|

y = |0-1|

y = | -1|

y = 1

The graph has exactly one -intercept, (0, 1) .

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.

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Example Question #20 : Dsq: Calculating The Slope Of A Line

Give the slope of a line on the coordinate plane.

Statement 1: The line passes through the vertex of the parabola of the equation $y = x^{2} + 4$ .

Statement 2: The line passes through the -intercept of the parabola of the equation $y = x^{2} + 4$ .

Possible Answers:

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.

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

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The vertex of the parabola of the equation $y = ax^{2}+bx + c$ can be found by first taking $x = -\frac{b}{2a}$ , then substituting in the equation and solving for .

$y = x^{2} + 4$

$x = -\frac{0}{2 \cdot 1} = 0$

$y = x^{2} + 4 = 0^{2} + 4 = 0+4 = 4$

The vertex is the point (0, 4) . Since x = 0 , 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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GMAT Math : Other Lines

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Dsq: Calculating The Slope Of A Line

Example Question #12 : Dsq: Calculating The Slope Of A Line

Example Question #13 : Dsq: Calculating The Slope Of A Line

Example Question #21 : Other Lines

Example Question #15 : Dsq: Calculating The Slope Of A Line

Example Question #16 : Dsq: Calculating The Slope Of A Line

Example Question #17 : Dsq: Calculating The Slope Of A Line

Example Question #18 : Dsq: Calculating The Slope Of A Line

Example Question #61 : Lines

Example Question #20 : Dsq: Calculating The Slope Of A Line

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