DSQ: Graphing a point - GMAT Quantitative

Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation

,

which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information. By the same argument, Statement 2 is also insuffcient.

Now assume both statements to be true. The two statements together form a system of linear equations which can be solved using the elimination method:

Now, substitute back:

The point is , which has a positive -coordinate and a negative -coordinate and is consequently in Quadrant IV.

Assume Statement 1 alone. The points and each satisfy the condition of the statement; however, the former is in Quadrant I, having a positive -coordinate and a positive -coordinate; the latter is in Quadrant IV, having a positive -coordinate and a negative -coordinate.

Assume Statement 2 alone. The points and each satisfy the condition of the statement, since . However, the former is in Quadrant IV, having a positive -coordinate and a negative -coordinate; the latter is in Quadrant II, having a negative -coordinate and a positive -coordinate.

Assume both statements to be true. Statement 2 can be rewritten as ; since is positive from Statement 1, is negative. Since the point has a positive -coordinate and a negative -coordinate, it is in Quadrant IV.

Two points in the same quadrant have -coordinates of the same sign and -coordinates of the same sign.

It is possible for two points fitting the condition of Statement 1 to be in the same quadrant; and are two such points. However, it is also possible for two such points to be in different quadrants; and are two such points. Therefore, Statement 1 alone gives insufficient information. By the same argument, Statement 2 alone gives insufficient information.

Assume both statements are true. and are of different sign by Statement 1. By Statement 2, and are of the same sign; therefore, they are both of the same sign as and the sign opposite that of , or vice versa. Therefore, in one ordered pair, both numbers are positive or both are negative, and in the other ordered pair, one number is positive and the other is negative. The two ordered pairs cannot represent points in the same quadrant.

Assume Statement 1 alone. The equations can be rewritten as follows:

The - and -coordinates of are the arithmetic means of those of and , so is the midpoint of the segment with those endpoints. Therefore, the three points are collinear.

Assume Statement 2 alone. The statement can be rewritten as follows:

The first expression is the slope of the line through and ; the second expression is the slope of the line through and . Since the slopes are equal, the three points are collinear.

Two points in the same quadrant have -coordinates of the same sign and -coordinates of the same sign; however, from Statement 1 alone, we find that the -coordinates of the points have different signs, and from Statement 2 alone, we find that this holds for the -coordinates. Therefore, from either statement alone, the points can be proved to be in different quadrants.

Assume both statements. The points and each satisfy the conditions of both statements, since , , and . The former is in Quadrant I, having a positive -coordinate and a positive -coordinate; the latter is in Quadrant IV, having a positive -coordinate and a negative -coordinate.

Assume Statement 1 alone.

The proportion statement

can be rewritten by setting the reciprocals of the expressions equal:

The first expression is the slope of the line through and ; the second is the slope of the line through and . Since the slopes are equal, the three points are on the same line - collinear.

The three points cannot be assumed to be collinear from Statement 2 alone. For example, , , and collectively fit the condition of Statement 2, and all three points are easily seen to be on the line of the equation . However, , , and collectively fit the condition of Statement 2, and while the line through the first two points is again , is off that line, so the three points are noncollinear.

Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation

which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information.

Now assume Statement 2 alone. The set of points that satisfy the equation is the set of all points of the circle of the equation

This circle has as its center and as its radius. Since its center is , which is 5 units away from its closest axis, and the radius is less than 5 units, the circle never intersects an axis, so it is contained entirely within the same quadrant as its center. The center has negative - and -coordinates, placing it, and the entire circle, in Quadrant III.

Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation

,

which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information. By the same argument, Statement 2 is also insuffcient.

Now assume both statements to be true. The two statements together form a system of linear equations which can be solved using the elimination method:

Now, substitute back:

The point is , which has a positive -coordinate and a negative -coordinate and is consequently in Quadrant IV.

Assume Statement 1 alone. The points and each satisfy the condition of the statement; however, the former is in Quadrant I, having a positive -coordinate and a positive -coordinate; the latter is in Quadrant IV, having a positive -coordinate and a negative -coordinate.

Assume Statement 2 alone. The points and each satisfy the condition of the statement, since . However, the former is in Quadrant IV, having a positive -coordinate and a negative -coordinate; the latter is in Quadrant II, having a negative -coordinate and a positive -coordinate.

Assume both statements to be true. Statement 2 can be rewritten as ; since is positive from Statement 1, is negative. Since the point has a positive -coordinate and a negative -coordinate, it is in Quadrant IV.

Two points in the same quadrant have -coordinates of the same sign and -coordinates of the same sign.

It is possible for two points fitting the condition of Statement 1 to be in the same quadrant; and are two such points. However, it is also possible for two such points to be in different quadrants; and are two such points. Therefore, Statement 1 alone gives insufficient information. By the same argument, Statement 2 alone gives insufficient information.

Assume both statements are true. and are of different sign by Statement 1. By Statement 2, and are of the same sign; therefore, they are both of the same sign as and the sign opposite that of , or vice versa. Therefore, in one ordered pair, both numbers are positive or both are negative, and in the other ordered pair, one number is positive and the other is negative. The two ordered pairs cannot represent points in the same quadrant.

Assume Statement 1 alone. The equations can be rewritten as follows:

The - and -coordinates of are the arithmetic means of those of and , so is the midpoint of the segment with those endpoints. Therefore, the three points are collinear.

Assume Statement 2 alone. The statement can be rewritten as follows:

The first expression is the slope of the line through and ; the second expression is the slope of the line through and . Since the slopes are equal, the three points are collinear.

Two points in the same quadrant have -coordinates of the same sign and -coordinates of the same sign; however, from Statement 1 alone, we find that the -coordinates of the points have different signs, and from Statement 2 alone, we find that this holds for the -coordinates. Therefore, from either statement alone, the points can be proved to be in different quadrants.

Assume both statements. The points and each satisfy the conditions of both statements, since , , and . The former is in Quadrant I, having a positive -coordinate and a positive -coordinate; the latter is in Quadrant IV, having a positive -coordinate and a negative -coordinate.

Assume Statement 1 alone.

The proportion statement

can be rewritten by setting the reciprocals of the expressions equal:

The first expression is the slope of the line through and ; the second is the slope of the line through and . Since the slopes are equal, the three points are on the same line - collinear.

The three points cannot be assumed to be collinear from Statement 2 alone. For example, , , and collectively fit the condition of Statement 2, and all three points are easily seen to be on the line of the equation . However, , , and collectively fit the condition of Statement 2, and while the line through the first two points is again , is off that line, so the three points are noncollinear.

Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation

which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information.

Now assume Statement 2 alone. The set of points that satisfy the equation is the set of all points of the circle of the equation

This circle has as its center and as its radius. Since its center is , which is 5 units away from its closest axis, and the radius is less than 5 units, the circle never intersects an axis, so it is contained entirely within the same quadrant as its center. The center has negative - and -coordinates, placing it, and the entire circle, in Quadrant III.

Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation

,

which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information. By the same argument, Statement 2 is also insuffcient.

Now assume both statements to be true. The two statements together form a system of linear equations which can be solved using the elimination method:

Now, substitute back:

The point is , which has a positive -coordinate and a negative -coordinate and is consequently in Quadrant IV.

Assume Statement 1 alone. The points and each satisfy the condition of the statement; however, the former is in Quadrant I, having a positive -coordinate and a positive -coordinate; the latter is in Quadrant IV, having a positive -coordinate and a negative -coordinate.

Assume Statement 2 alone. The points and each satisfy the condition of the statement, since . However, the former is in Quadrant IV, having a positive -coordinate and a negative -coordinate; the latter is in Quadrant II, having a negative -coordinate and a positive -coordinate.

Assume both statements to be true. Statement 2 can be rewritten as ; since is positive from Statement 1, is negative. Since the point has a positive -coordinate and a negative -coordinate, it is in Quadrant IV.

Two points in the same quadrant have -coordinates of the same sign and -coordinates of the same sign.

It is possible for two points fitting the condition of Statement 1 to be in the same quadrant; and are two such points. However, it is also possible for two such points to be in different quadrants; and are two such points. Therefore, Statement 1 alone gives insufficient information. By the same argument, Statement 2 alone gives insufficient information.

Assume both statements are true. and are of different sign by Statement 1. By Statement 2, and are of the same sign; therefore, they are both of the same sign as and the sign opposite that of , or vice versa. Therefore, in one ordered pair, both numbers are positive or both are negative, and in the other ordered pair, one number is positive and the other is negative. The two ordered pairs cannot represent points in the same quadrant.