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.

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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\( = 6\).

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Attachment:

ParallelogramGraph.jpg [ 29.6 KiB | Viewed 6799 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

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

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

Prep Toolkit

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My Rewards