Coordinate Geometry - GMAT Quantitative

Statement 1: Since we're referring to a line parallel to line XY, the slopes will be identical. We can use the points provided to calculate the slope:

We can simplify the slope to just .

Statement 2: Finding the slope of a line parallel to line XY is really straightforward when given the equation of a line.

Where is the slope and the y-intercept.

In this case, our value is .\

Each statement alone is sufficient to answer the question.

Recall that parallel lines have the same slope and that slope can be calculated from any two points.

Statement I gives us a point on

Statement II gives us the x-intercept, a.k.a. the point .

Therfore, using both statements, we can find the slope of and any line parallel to it.

The lines are parallel, so their slopes are equal.

The slope of the first is .

The slope of the second is .

Set the two equal to each other:

If you know that , then you can easily find by substituting:

Cross-multiply and solve:

If you know that , do the same thing:

Therefore, either statement alone is sufficient to answer the question.

Two parallel lines must have the same slope. Therefore, the product of the slopes will be the product of two real numbers of like sign, which must be positive. Each of the two statements contradicts this, so either statement alone answers the question.

Assume both statements are true. Then three things are possible:

Case 1: Both lines will have slope 1, or

Case 2: Both lines will have slope

In either case, since the lines have the same slope, they are parallel.

Case 3: One line has slope 1 and one has slope

In this case the lines are perpendicular.

The two statements therefore provide insufficient information.

The question can be answered by finding and comparing the slopes of the lines. The lines are parallel if and only if they have the same slope, and perpendicular if and only if the product of the slopes is .

Statement 1 alone does not answer the question. Two lines with slope 1 are parallel, and a line with slope 2 and a line with slope are not, but in both cases, the product of the slopes is 1.

Statement 2 alone gives that Line has slope 1 or , but nothing is given about the slope of Line .

Now, assume both statements are true. From Statement 2, has slope 1 or . From Statement 1, the product of the slopes is 1; if the slope of is 1, then the slope of is , and if the slope of is , then the slope of is . Therefore, if both statements are true, the lines have the same slope, making them parallel.

The answer to the question depends on the slopes of the lines - parallel lines have the same slope, and perpendicular lines have slopes that have product .

Statement 1 alone only eliminates the possiblity of the lines being perpendicular, since the product of the slope is not . Two lines with slope 3 are parallel, and one line with slope 1 and one with slope 9 are neither parallel nor perpendicular; both pairs of lines satisfy Statement 1, but only the first pair is parallel. Therefore, Statement 1 only establishes that they are not perpendicular.

From Statement 2, only the slope of is given; without the slope of , the question cannot be answered.

Assume both statements to be true. Then since Line has slope and the product of the slopes is 9, The slope of Line is . Therefore, both lines have slope , and the lines are parallel.

The answer to the question depends on the slopes of the lines—parallel lines have the same slope, and perpendicular lines have slopes that have product .

From Statement 1 alone, we only know one point of each line, so no information about their slopes can be determined; the same holds for Statement 2.

Assume both statements are true. Then we know two points of each line—specifically, both intercepts—from which we can determine the slope of each by way of the slope formula. After doing so, we can use the slopes to answer the question.

Two lines are parallel if and only if they have the same slope, and perpendicular if and only if their slopes have product .

Assume Statement 1 alone. Since the product of the slopes is , the lines are perpendicular.

Statement 2 alone is unhelpful, since no information about the slope of a line can be determined from only one point.

Two lines can be determined to be parallel, perpendicular, or neither from their slopes.

Assume Statement 1 alone. Parallel lines must have the same slope, so this choice can be eliminated. The slopes of perpendicular lines must have product ; since the product of the slopes is , this choice can be eliminated as well. It can therefore be deduced that the lines are neither parallel nor perpendicular.

Assume Statement 2 alone. Since the lines have at least one point in common, they are not parallel, but this is the only choice that can be eliminated.

Two lines can be determined to be parallel, perpendicular, or neither from their slopes.

Assume Statement 1 alone is true. Since these distinct lines have the same slope, they are parallel.

Assume Statement 2 alone is true. No information about the slopes of the lines can be determined from one single point, so Statement 2 alone is insufficient.

In this case, I and II give us a ton of information about the size of the circle, but we have no clue as to its location.

In order to know the slope of the tangent line, we need to know location of the circle, so we cannot solve this problem.

To find the equation of a linear function, we need some combination of slope and a point.

Statement I gives us a clue to find the slope of the desired function. It must be the opposite reciprocal of the slope of . This makes the slope of equal to

Statement II gives us a point on our desired function, .

Using slope-intercept form, we get the following:

So our equation is as follows

Find the equation for linear function p(x)

I) and

II)

To begin:

I) Tells us that p(x) must have a slope of 16

II) Tells us a point on p(x). Plug it in and solve for b:

Given that the square of a negative is still positive, it is possible for a to have an x-intercept that is negative, while still having a positive slope. The example above shows how the square of the x-intercept for line a could be greater, while having still giving line a a slope that is less than that of b.

Either statement is sufficient.

Line j, as a line, has an equation of the form

Statement 1 gives the equation of a perpindicular line, so the slopes of the two lines are negative reciprocals of each other:

Statement 2 allows the slope to be found using rise over run:

Then, since the x-intercept is known:

Assume both statements to be true. Then the line shares its - and -intercepts with the graph of , which is a parabola. The common -intercept can be found by setting and solving for :

,

making the -intercept of the parabola, and that of the line, .

The common -intercept can be found by setting and solving for :

, in which case , or

, in which case ,

The parabola therefore has two -intercepts, and , so it is not clear which one is the -intercept of the line. Therefore, the equation of the line is also unclear.

, in slope-intercept form, is

Therefore, the sign of is the sign of the slope.

The first statement means that is positive - all that means is that both and are nonzero and of like sign. can be either positive or negative, and consequently, so can slope .

The second statement - that is positive - makes , the sign of the slope, negative.

The slope of a line through and is

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