GMAT Math : Other Lines

Example Questions

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Example Question #1 : Dsq: Calculating Whether Point Is On A Line With An Equation

Determine whether the points are collinear.

Statement 1: The three points are

Statement 2: Slope of line and the slope of line

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

EACH statement ALONE is sufficient.

Explanation:

Points are collinear if they lie on the same line.  Here A, B, and C are collinear if the line AB is the same as the line AC.  In other words, the slopes of line AB and line AC must be the same.  Statement 2 gives us the two slopes, so we know that Statement 2 is sufficient.  Statement 1 also gives us all of the information we need, however, because we can easily find the slopes from the vertices.  Therefore both statements alone are sufficient.

Example Question #2 : Dsq: Calculating Whether Point Is On A Line With An Equation

Given:

Find .

I) .

II) The  coordinate of the minmum of  is .

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Explanation:

By using I) we know that the given point is on the line of the equation.

So I) is sufficient.

II) gives us the y coordinate of the minimum. In a quadratic equation, this is what  "c" represents.

Therefore, c=-80 and II) is also sufficient.

Example Question #3 : Dsq: Calculating Whether Point Is On A Line With An Equation

Find whether the point  is on the line .

I)  is modeled by the following: .

II)  is equal to five more than 3 times the y-intercept of .

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Both statements are needed to answer the question.

Both statements are needed to answer the question.

Explanation:

To find out if a point is on line with an equation, one can simply plug in the point; however, this is complicated here by the fact that we are missing the x-coordinate.

Statement I gives us our function.

Statement II gives us a clue to find the value of .  is five more than 3 times the y-intercept of . So, we can find the following:

To see if the point  is on the line , plug it into the function:

This is not a true statement, so the point is not on the line.

Example Question #4 : Dsq: Calculating Whether Point Is On A Line With An Equation

Consider linear functions  and

I)  at the point .

II)

Is the point  on the line ?

Both statements are needed to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Explanation:

Consider linear functions h(t) and g(t).

I)  at the point

II)

Is the point  on the line h(t)?

We can use II) and I) to find the slope of h(t)

Recall that perpendicular lines have opposite reciprocal slope. Thus, the slope of h(t) must be

Next, we know that h(t) must pass through (6,4), so lets us that to find the y-intercept:

Next, check if (10,4) is on h(t) by plugging it in.

So, the point is not on the line, and we needed both statements to know.

Example Question #5 : Dsq: Calculating Whether Point Is On A Line With An Equation

Line m is perpendicular to the line l which is defined by the equation . What is the value of ?

(1) Line m passes through the point .

(2) Line l passes through the point .

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Explanation:

Statement 1 allows you to define the equation of line m, but does not provide enough information to solve for .  There are still 3 variables  and only two different equations to solve.

if , statement 2 supplies enough information to solve for b by substitution if  is on the line.

Example Question #1 : Dsq: Calculating The Equation Of A Line

Find the equation of linear function  given the following statements.

I)

II)  intercepts the x-axis at 9.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Explanation:

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

Example Question #2 : Dsq: Calculating The Equation Of A Line

There are two lines in the xy-coordinate plane, a and b, both with positive slopes. Is the slope of a greater than the slope of b?

1)The square of the x-intercept of a is greater than the square of the x-intercept of b.

2) Lines a and b have an intersection at

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Neither of the statements, together or separate, is sufficient.

Together the two statements are sufficient.

Either of the statements is sufficient.

Neither of the statements, together or separate, is sufficient.

Explanation:

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.

Example Question #3 : Dsq: Calculating The Equation Of A Line

Line j passes through the point . What is the equation of line j?

1) Line j is perpindicular to the line defined by

2) Line j has an x-intercept of

Neither of the statements, separate or together, is sufficient.

Statement 2 alone is sufficient.

Either of the statements is sufficient.

Statement 1 alone is sufficient.

Together, the two statements are sufficient.

Either of the statements is sufficient.

Explanation:

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:

Example Question #4 : Dsq: Calculating The Equation Of A Line

Find the equation for linear function .

I)  and

II)

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Both statements are needed to answer the question.

Explanation:

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:

Example Question #5 : Dsq: Calculating The Equation Of A Line

Give the equation of a line.

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

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

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation:

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.

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