What is the value of x?

(1) 310x + 562y = 909

(2) 951x - 626y = 323

OA is C

I am tempted to go for C but before that step I should work on both the statements independently but working on them can take a lot of time since they both are ugly looking equations. So, is there any other technique to solve for s(1) and s(2) independently without spending too much time ?

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Since the nature of y coefficients of the two equations is the opposite, (eqn (1)'s y coefficient is positive, while eqn (2)'s y coefficient is negative), it is for sure that you would be a unique value of x and y. Sufficient.

This is a DS question, you need to be sure that you get the answer; there is no need to calculate the answer.

Hope this helps!

-Jay

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(1) 310x + 562y = 909

We can think of this as the equation of a line.

The x- and y-coordinates of all points on the line will satisfy the equation 310x + 562y = 909

Since there are infinitely many points on the line, there are infinitely many possible values of x

INSUFFICIENT

(2) 951x - 626y = 323

If we think of this as the equation of a line, we can also draw the same conclusion that statement 2 is INSUFFICIENT

(1 & 2 combined)

There are two possible cases to consider:

case a) The two equations represent the SAME line.

In this case, there are still infinitely many points (x, y) that will satisfy both equations, in which case the combined statements are not sufficient.

case b) The two equations represent DIFFERENT lines.

In this case, the two lines will intersect at exactly 1 point, which means there will only one pair of values (x, y) that satisfy both equations, in which case the combined statements are sufficient.

So, which is it?

Do the two equations represent the same line or different lines?

To find out, let's examine the slope of each line by rewriting the equations in slope y-intercept form.

(1) 310x + 562y = 909

562y = - 310x + 909

y = (-310/562)x + 909/562

(2) 951x - 626y = 323

951x = 323 + 626y

951x - 323 = 626y

(951/626)x - 323/626 = y

or... y = (951/626)x - 323/626

So, the slope of the 1st line is -310/562, and the slope of the 2nd line is 951/626

Since the slopes are different (one is positive and one is negative), the two equations CANNOT represent the same line.

As such, the two lines must intersect at ONE point.

So, IF we were to solve the system, we'd find exactly one solution.

This means the combined solutions must be SUFFICIENT.

Does that help?

Cheers,

Brent

(1) Insufficient

Take y = 0 , hence ? = x = 909/310

Take y = 1, hence ? = x = (909-562)/310 , different from 909/310

(2) Insufficient (Same procedure)

(1+2) BrentÂ´s "lines argument" is important, because the sufficiency is immediate based on it:

The slopes are different: "line (1)" slope is negative (it is -310/562, without any need of writing) , the slope of "line (2)" is positive (951/626 , again no need of writing) ...

Conclusion: lines (1) and (2) are (coplanar and) not parallel (*), hence they are concurrent, that is, a unique point of intersection.

Hence the abscissa ("x") of the point of intersection is unique, and we are done: (1+2) Sufficient.

(*) Do not forget there are two "types" of parallelism: lines may be [coincident] or [parallels and distinct].

The above follows the notations and rationale taught in the GMATH method.

## What is the value of x?

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With such ugly-looking numbers, make sure that the given two equations are unique, i.e., they are two simultaneous equations guaranteeing unique values of x and y.vinni.k wrote:What is the value of x?

(1) 310x + 562y = 909

(2) 951x - 626y = 323

OA is C

I am tempted to go for C but before that step I should work on both the statements independently but working on them can take a lot of time since they both are ugly looking equations. So, is there any other technique to solve for s(1) and s(2) independently without spending too much time ?

Since the nature of y coefficients of the two equations is the opposite, (eqn (1)'s y coefficient is positive, while eqn (2)'s y coefficient is negative), it is for sure that you would be a unique value of x and y. Sufficient.

This is a DS question, you need to be sure that you get the answer; there is no need to calculate the answer.

Hope this helps!

-Jay

_________________

Manhattan Review

Locations: Manhattan Review Kukatpally | GMAT Prep Jayanagar | GRE Prep Tarnaka | Madhapur GRE Coaching | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.

- vinni.k
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Thanks but i am not sure how did you decide to go for C. If nature of y coefficients of the two equations is opposite, then there will be unique value of x and y. Is this the reason for not attempting s(1) and s(2) independently ?

Can you please elaborate more on this part ? I am just interested in how to avoid working on s(1) and s(2) independently because in the exam if somehow i ended up making an ugly looking equation, then i might have to solve it, and it will take a hell of a time.

Can you please elaborate more on this part ? I am just interested in how to avoid working on s(1) and s(2) independently because in the exam if somehow i ended up making an ugly looking equation, then i might have to solve it, and it will take a hell of a time.

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Here's another way to look at it.....vinni.k wrote:What is the value of x?

(1) 310x + 562y = 909

(2) 951x - 626y = 323

(1) 310x + 562y = 909

We can think of this as the equation of a line.

The x- and y-coordinates of all points on the line will satisfy the equation 310x + 562y = 909

Since there are infinitely many points on the line, there are infinitely many possible values of x

INSUFFICIENT

(2) 951x - 626y = 323

If we think of this as the equation of a line, we can also draw the same conclusion that statement 2 is INSUFFICIENT

(1 & 2 combined)

There are two possible cases to consider:

case a) The two equations represent the SAME line.

In this case, there are still infinitely many points (x, y) that will satisfy both equations, in which case the combined statements are not sufficient.

case b) The two equations represent DIFFERENT lines.

In this case, the two lines will intersect at exactly 1 point, which means there will only one pair of values (x, y) that satisfy both equations, in which case the combined statements are sufficient.

So, which is it?

Do the two equations represent the same line or different lines?

To find out, let's examine the slope of each line by rewriting the equations in slope y-intercept form.

(1) 310x + 562y = 909

562y = - 310x + 909

y = (-310/562)x + 909/562

(2) 951x - 626y = 323

951x = 323 + 626y

951x - 323 = 626y

(951/626)x - 323/626 = y

or... y = (951/626)x - 323/626

So, the slope of the 1st line is -310/562, and the slope of the 2nd line is 951/626

Since the slopes are different (one is positive and one is negative), the two equations CANNOT represent the same line.

As such, the two lines must intersect at ONE point.

So, IF we were to solve the system, we'd find exactly one solution.

This means the combined solutions must be SUFFICIENT.

Does that help?

Cheers,

Brent

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? = xvinni.k wrote:What is the value of x?

(1) 310x + 562y = 909

(2) 951x - 626y = 323

(1) Insufficient

Take y = 0 , hence ? = x = 909/310

Take y = 1, hence ? = x = (909-562)/310 , different from 909/310

(2) Insufficient (Same procedure)

(1+2) BrentÂ´s "lines argument" is important, because the sufficiency is immediate based on it:

The slopes are different: "line (1)" slope is negative (it is -310/562, without any need of writing) , the slope of "line (2)" is positive (951/626 , again no need of writing) ...

Conclusion: lines (1) and (2) are (coplanar and) not parallel (*), hence they are concurrent, that is, a unique point of intersection.

Hence the abscissa ("x") of the point of intersection is unique, and we are done: (1+2) Sufficient.

(*) Do not forget there are two "types" of parallelism: lines may be [coincident] or [parallels and distinct].

The above follows the notations and rationale taught in the GMATH method.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)

English-speakers :: https://www.gmath.net

Portuguese-speakers :: https://www.gmath.com.br

English-speakers :: https://www.gmath.net

Portuguese-speakers :: https://www.gmath.com.br

- vinni.k
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Brent and Fabio,

Thanks for your response. The above is a very good way of solving this question with coordinate geometry, but the one thing that I forgot in this question is that it is not given that x is an integer. So, x can have infinitely many values. However, it's a nice way of thinking this question in slope intercept form.

Thanks...really appreciate it <i class="em em-blush"></i>

Thanks for your response. The above is a very good way of solving this question with coordinate geometry, but the one thing that I forgot in this question is that it is not given that x is an integer. So, x can have infinitely many values. However, it's a nice way of thinking this question in slope intercept form.

Thanks...really appreciate it <i class="em em-blush"></i>

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Ops... I clicked "+1 Upvote Post" on vinni.k Â´s last comment because I thought it was something like the "like" in facebook... I am sorry.... I am just beginning to use the forum!

Pressed the same button in BrentÂ´s answer so that it would go to the top again. It worked!

Pressed the same button in BrentÂ´s answer so that it would go to the top again. It worked!

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)

English-speakers :: https://www.gmath.net

Portuguese-speakers :: https://www.gmath.com.br

English-speakers :: https://www.gmath.net

Portuguese-speakers :: https://www.gmath.com.br