in the standard (xy) coordinate plane the graph of 3x+7y

Question

In the standard (x,y) coordinate plane, the graph of \(|3x| + |7y| = 21\) is a parallelogram with 2 vertices on the x-axis and 2 vertices on the y-axis. Let H and V be the lengths, in coordinate units, of the horizontal and vertical diagonals, respectively, of this parallelogram.

Select for H and for V values that are consistent with the given information. Make only two selections, one in each column.

Show Spoiler

Attachment:

DI -3.png

chetan2u · Accepted Answer

The question tells us that the points make a parallelogram. 
Even the solution of equation will give two intersection on x axis at equal distance from origin and similarly 2 points on y axis at equal distance from origin. 
One of the diagonals will be formed by y-axis intersection and the other by x-axis intersection. 
Let us solve it

|3x| + |7y| = 21  Take y as 0 to find intersection at y-axis. |3x|=21 or x=7 and -7.
Horizontal Diagonal = 7-(-7)=14
 |3x| + |7y| = 21  Take x as 0 to find intersection at x-axis. |7y|=21 or y=3 and -3.
Vertical Diagonal = 3-(-3)=6

sukoon9334 , you are correct with your method.

sukoon9334 · Answer

It's an 80% inequalities question and 20% coordinate planes.

|3x|+|7y|=21 can be written as |7y|=21-|3x|
Now, since |7y| is always zero or positive, hence, 21 - |3x| will also always be zero or positive.

This means that 0≤ |3x|≤ 21 
Case 1: 3x ≤ 21
=> x≤7

Case 2: -3x≤21
=> x≥-7

Case 3: 0≤ |3x|
=> x=0

Analysis I: 
|7y|=0 for both extremes i.e. x=7 and x=-7
This implies, y=0
So, our two coordinate points of the parallelogram are (7,0) and (-7,0)

Analysis II:
|7y|=21 for x=0

This creates two more cases:
Case 3: 7y≤ 21
=> y≤ 3

Case 4: -7y≤ 21
=> y≥ -3
So, our other two coordinates for the parallelogram are (0,3) and (0,-3)

Now, when we plot these 4 coordinates on the coordinate plane, we get a shape whose diagonals are the x-axis and y-axis itself. 
V=distance from (0,3) to (0,-3) = 6 units
H= distance from (7,0) to (-7,0) = 14 units

This might seem daunting. I felt the same when I saw it and got it incorrect in the mock test but when I sat down later, I got this one in one shot. All the best.

Bunuel  approve this, please.

JoeAa · Answer

Isn't geometry not being tested? why is it being asked here?

I got this wrong because I am trying to imagine how'd a parallelogram have all vertices lied on both X and Y axsis.

chetan2u · Answer

Exactly, there is no way a parallelogram can be made using x axis and y axis unless opposite vertices lie on the same axis. 
Just substitute x=0, to get those y-coordinates, and vice versa.

ruis · Answer

Can you please add the tag Coordinate Geometry please? Thank you

sayan640 · Answer

PFA the grpahical representation of this problem.  GMATinsight   chetan2u

MintTea · Answer

Here is my approach, take around 2mins

We have 4 situations:
3x + 7y = 21 (1)
3x - 7y = 21 (2)
-3x +7y =21 (3)
-3x - 7y = 21 (4)

So, let sum (1) and (2) => 6x = 42, => x= 7
let sum (1) and (3) => 14y = 42 => y =3

Double both get the answer H = 2x = 14, and V = 2y = 6

Bismuth83 · Answer

1. It's always better to visualize the problem: the parallelogram, H, and V. We are asked to find H and V.

2. Note that the figure must be centered at (0,0) since if (x,y) is a solution, then so is (-x,y), (x,-y), and (-x,-y).

3. To find H and V we need to find the coordinates of the vertices. Let them be (x,0), (-x,0), (0,y), and (0,-y). Also let x and y be positive.

4. Now let's add the condition that each of these points must work in the equation: \(|3x| + |7y| = 21\).

(x,0). \(|3x| + |7y| = 21 \rightarrow 21 = |3 * x| + |7 * 0| = |3x| \rightarrow x = \pm 7\). Since x > 0, x = 7 and the point is (7,0).

(-x,0). This case is the same as (x,0) but the point will be (-7,0).

(0,y). \(|3x| + |7y| = 21 \rightarrow 21 = |3 * 0| + |7 * y| = |7y| \rightarrow y = \pm 3\). Since y > 0, y = 3 and the point is (0,3).

(0,-y). This case is the same as (0,y) but the point will be (0,-3).

5. Based on the image, H = 2x and V = 2y. So, H\( = 14\) and V\( = 6\).

Show Spoiler

Attachment:

ParallelogramGraph.jpg [ 29.6 KiB | Viewed 6198 times ]

KarishmaB · Answer

The question involves some graphing knowledge, some basic knowledge about shapes (what is a parallelogram? As long as you know it is a 4 sides closed figure, that's enough) and some basic absolute values concepts. Overall, it is typical GMAT question - involves multiple areas but is simple. Graph of |x| + |y| = a is a closed 4 sided figure around (0, 0) with diagonals on X and Y axis. I discussed graphing this in this video on YouTube: https://youtu.be/dPwGi8oV9ns Even if we do not know this, we are given that 2 vertices are on X axis and 2 are on Y axis and H and V are the diagonals. Then the diagonals must be on X and Y axis. All we need to do is find the points on the axis to get the lengths of the diagonals. |3x| + |7y| = 21 On Y axis, x = 0 so |7y| = 21 y = 3 or -3 Length of vertical diagonal V = 6 On X axis, y = 0 so |3x| = 21 x = 7 or -7 Length of horizontal diagonal H = 14

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

in the standard (xy) coordinate plane the graph of 3x+7y

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Two-part Analysis (TPA) Questions

in the standard (xy) coordinate plane the graph of 3x+7y

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards