Lines - GMAT Quantitative Reasoning

Find endpoint Y given the following:

I) Segment RY has its midpoint at (45,65)

II) Point R is located on the x-axis, 13 points from the origin.

I) Gives us the location of the midpoint of our segment

(45,65)

II) Gives us the location of one endpoint

(13,0)

Use I) and II) to work backwards with midpoint formula to find the other endpoint.

So endpoint is at .

Therefore, both statements are needed to answer the question.

To find the endpoint of a segment, we can generally use the midpoint formula; however, in this case we do not have enough information.

I) Gives us one endpoint

II) Gives us the length of DF

The problem is that we don't know the orientation of DF. It could go in infinitely many directions, so we can't find the location of without more information.

The midpoint formula is

,

with statement 1, we know that Q is and can solve for P:

and

Statement 1 alone is sufficient.

Statement 2 doesn't provide enough information to solve for point P.

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 order to calculate the equation of a line that passes through a point, we need one of two pieces of information. If we know another point, we can calculate the slope and solve for the -intercept, giving us the equation of the line. Alternatively, if we know the slope (which we can conclude from the parallel line in statement 2) we can calculate the -intercept and determine the equation of the line.

To find the equation of a parallel line, we need the slope and the y-intercept.

Parallel lines have the same slope, so we have that.

I and II each give us a point on the graph, so we could find the equation of the line through either of them.

Statement 1: We're given the slope line AB, because we are ask for the equation of the line we need more than just the slope of the line. Therefore, this information alone is not sufficient to write an actual equation.

Statement 2: Using the information from statement 1 and the points provided in this statement, we can answer the question.

We are asked to find the equation of a line related to another line.

Statement I tells us the two lines are parallel. This means they have the same slope

Statement II gives us a point on our desired line. We can use this to find the line's y-intercept, which will then allow us to write its equation.

Plug all of the given info into slope-intercept form and solve for b, the line's y-intercept:

So our equation is:

To find the equation of any line, we need 2 pieces of information, the slope of the line and any point on the line. From statement 1, we get a point on Line AB. From statement 2, we get the slope of Line BC. Since we know that AB is perpindicular to BC, we can derive the slope of AB from the slope of BC. Therefore to find the equation of the line, we need the information from both statements.

To find the equation of a perpendicular line you need the slope of the line and a point on the line. We can find the slope by knowing g(x).

I) Gives us a point on h(x).

II) Gives us the y-intercept of h(x).

Either of these will be sufficient to find the rest of our equation.

Statement 1: Since the line we're looking for is perpendicular to line XY, our slope will be the inverse of line XY's slope .

The slope of our line is then . Just knowing the slope however, is not sufficient information to answer the question.

Statement 2: We're provided with a point which will allow us the write the equation.

Statement 1: We're given the equation to line AB which contains the slope. Because the line we're being asked for is perpendicular to it, we know the slope will be its inverse.

The slope of our line is then

Statement 2: We can write the equation to the perpendicular line only if we have a point that falls within that line. Luckily, we're given such a point in statement 2.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Find the equation of the line perpendicular to r(x)

I) r(x) has a slope of -15

II) The line must pass through the point (9, 96)

Recall that perpendicular lines have opposite reciprocal slopes.

Use I) to find the slope of our new line

Use II) along with our slope to find the y-intercept of our new line.

Therefore both statements are needed.

Recall that perpendicular lines have opposite, reciprocal slopes. We can find the slope of from the question.

Statement I gives us a point on , which we can use to find the y-intercept of , and then the equation.

The slope of must be the opposite reciprocal of , this makes our slope .

Statement I tells us that passes through the point , so we can use slope-intercept form to find our equation:

So, our equation is

Statement II gives us a point on , which does not help us in the slightest with . Therefore, only Statement I is sufficient.

Assume Statement 1 alone. The -intercept of the line of the equation can be found by substituting and solving for :

The -intercept of the line is the origin ; it follows that this is also the -intercept.

Therefore, Statement 1 alone yields only one point of the line, from which its equation cannot be determined.

Assume Statement 2 alone. The slope of the line of the equation can be calculated by putting it in slope-intercept form :

The slope of this line is the coefficient of , which is . A line perpendicular to this one has as its slope the opposite of the reciprocal of , which is

.

However, there are infintely many lines with this slope, so no further information can be determined.

Now assume both statements to be true. From Statement 1, the slope of the line is , and from Statement 2, the -coordinate of the -initercept is . Substitute in the slope-intercept form: