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In the rectangular coordinate system, a line passes through the points

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In the rectangular coordinate system, a line passes through the points [#permalink] New post   25 Nov 2011, 16:59
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In the rectangular coordinate system, a line passes through the points (0,5) and (7,0). Which of the following points must the line also pass through?

A)(-14, 10)

B)(-7, 5)

C)(12, -4)

D)(14, -5)

E)(21, -9)

can someone please suggest a strategy for this type of problems?

thanks alot
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   26 Nov 2011, 07:21
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The conventional approach here would be to find the equation of the line. You can find the slope by subtracting the y coordinates of the two points, and then dividing by the difference in x coordinates:

m = (0 - 5)/(7 - 0) = -5/7

The y intercept of the line is 5, since (0,5) is on the line, so the equation of the line is

y = (-5/7)x + 5

If a point is on this line, then the equation above must be true if we plug in that point's coordinates. So we can now plug in each answer choice to see which works. If you plug in answer D (x = 14, y = -5), you get:

-5 = (-5/7)(14) + 5
-5 = -10 + 5
-5 = -5

so the equation is true, and D is on the line.

If you have a good conceptual understanding of slopes, you can bypass the calculations above. If a line travels from (0,5) to (7,0), it goes across 7 units and falls 5 units. So if we go across a further 7 units from the point (7,0), we must fall by a further 5 units. We'd then be at point (14,-5).
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   11 Feb 2019, 02:28
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My solution.

Use the two points formula for slopes:

y-y1/y2-y1 = x-x1/x2-x1

substitute the points x1=0,y1=5,x2=7,y2 =0

On solving, you will get the line equation 5x + 7y = 35

Now, start plugging in the answer choices to see which answer choice matches equation. Only option D does.

5(14) + 7(-5)= 35

LHS=RHS

Ans D.

Give a kudos if this explanation helped you :angel: .
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   11 Feb 2019, 03:01
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bmahes wrote:
IanStewart wrote:
The conventional approach here would be to find the equation of the line. You can find the slope by subtracting the y coordinates of the two points, and then dividing by the difference in x coordinates:

m = (0 - 5)/(7 - 0) = -5/7

The y intercept of the line is 5, since (0,5) is on the line, so the equation of the line is

y = (-5/7)x + 5

If a point is on this line, then the equation above must be true if we plug in that point's coordinates. So we can now plug in each answer choice to see which works. If you plug in answer D (x = 14, y = -5), you get:

-5 = (-5/7)(14) + 5
-5 = -10 + 5
-5 = -5

so the equation is true, and D is on the line.

If you have a good conceptual understanding of slopes, you can bypass the calculations above. If a line travels from (0,5) to (7,0), it goes across 7 units and falls 5 units. So if we go across a further 7 units from the point (7,0), we must fall by a further 5 units. We'd then be at point (14,-5).

So I wanted to get this "good conceptual understanding of slopes" and not solve it by making equations. Can you please help me with it?



I have discussed the concept of slope here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2016/0 ... line-gmat/

The points on the line are (0, 5) and (7, 0). So for 7 units increase in x value, y decreases by 5 units.

If x reduces by 7 units, y will increase by 5 units. So from (0, 5) we will get that (-7, 10) will lie on the line, not (-7, 5)
If x increases by 14 units, y will reduce by 10 units so from (0, 5), we will get (14, -5). This is option (D). Correct answer.
Some more examples:
If x reduces by 14 units, y will increase by 10 units so from (0, 5), we will get (-14, 15), not (-14, 10)
If x increases by 21 units, y will reduce by 15 units so from (0, 5), we will get (21, -10), not (21, -9)
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   27 Mar 2020, 05:10
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manalq8 wrote:
In the rectangular coordinate system, a line passes through the points (0,5) and (7,0). Which of the following points must the line also pass through?

A)(-14, 10)

B)(-7, 5)

C)(12, -4)

D)(14, -5)

E)(21, -9)

can someone please suggest a strategy for this type of problems?

thanks alot


We see that the slope of the line is (0 - 5) / (7 - 0) = -5/7. Now let’s check each given answer choice by determining the slope between that point and (0, 5).

A. (10 - 5) / (-14 - 0) = 5/(-14) = -5/14 → This is not -5/7.

B. (5 - 5) / (-7 - 0) = 0/(-7) = 0 → This is not -5/7.

C. (-4 - 5) / (12 - 0) = -9/12 = -3/4 → This is not -5/7.

D. (-5 - 5) / (14 - 0) = -10/14 = -5/7 → This is -5/7.

Answer: D
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   23 Mar 2015, 15:14
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we can from the start eliminate C and E, since the coordinate of y is clearly not divisible by 5. why 5? because the slope of the line is -(5/7)

thus we are left with
A)(-14, 10)
B)(-7, 5)
D)(14, -5)

we can further eliminate B, since this point will give a numerator equal to 0 when trying to plug in and check whether the slope is the same.

10-15 seconds was needed to eliminate 3 answer choices. Now we are left with only 2.
let's take A
we know that the slope of the line is -5/7
we can then check:
10 -5 = 5
-14 - 0 = -14
we get a slope of -(5/14)

this is clearly not what we need.

just to make sure, let's plug in values of the second given point
10-0 = 10
-14 - 7 = -21

we then get -(10/21)
we can then conclude that A is not the right answer.

The only answer choice left is D. which must be the right one.
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   19 Mar 2020, 10:37
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manalq8 wrote:
In the rectangular coordinate system, a line passes through the points (0,5) and (7,0). Which of the following points must the line also pass through?

A)(-14, 10)
B)(-7, 5)
C)(12, -4)
D)(14, -5)
E)(21, -9)


Here's another possible approach..

First plot of the two given points:


Notice that, to get from one point (0, 5) to the other point (7, 0), we must go down 5 units and then to the right 7 units


If we continue this pattern....

.... The next point on the line is at (14, -5)

Answer: D

Cheers,
Brent
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   13 Apr 2020, 02:51
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Kritisood wrote:

I have read your blog (highly helpful) and have understood that if the slope here were +2 it would mean if x increases by 1 unit, y would increase by 2 units. In this question, the slope is -5/7 (a fraction) now how do we apply this logic here? How did you come to this conclusion "for 7 units increase in x value, y decreases by 5 units"


Slope is 'change in y' for every 1 unit 'change in x'.

Slope = -5/7

Change in y = -5/7 for change in x = 1

So if change in x is 7, change in y = -5/7 * 7 = -5

So when x increases by 7, y decreases by 5 (that's what the negative sign shows that the change in y is opposite)
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   09 Apr 2016, 07:53
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manalq8 wrote:
In the rectangular coordinate system, a line passes through the points (0,5) and (7,0). Which of the following points must the line also pass through?
A)(-14, 10)
B)(-7, 5)
C)(12, -4)
D)(14, -5)
E)(21, -9)

can someone please suggest a strategy for this type of problems?

thanks alot



Hey :)

The first thing I always do is trying to draw this as fast and accurate as possible.
When you draw the two points in a coordinate system, connect them with a line and extend the line, you clearly see what point could be on the line.
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In the rectangular coordinate system, a line passes through the points [#permalink] New post   11 Feb 2019, 02:14
IanStewart wrote:
The conventional approach here would be to find the equation of the line. You can find the slope by subtracting the y coordinates of the two points, and then dividing by the difference in x coordinates:

m = (0 - 5)/(7 - 0) = -5/7

The y intercept of the line is 5, since (0,5) is on the line, so the equation of the line is

y = (-5/7)x + 5

If a point is on this line, then the equation above must be true if we plug in that point's coordinates. So we can now plug in each answer choice to see which works. If you plug in answer D (x = 14, y = -5), you get:

-5 = (-5/7)(14) + 5
-5 = -10 + 5
-5 = -5

so the equation is true, and D is on the line.

If you have a good conceptual understanding of slopes, you can bypass the calculations above. If a line travels from (0,5) to (7,0), it goes across 7 units and falls 5 units. So if we go across a further 7 units from the point (7,0), we must fall by a further 5 units. We'd then be at point (14,-5).


So I wanted to get this "good conceptual understanding of slopes" and not solve it by making equations. Can you please help me with it?
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   10 Apr 2020, 02:58
VeritasKarishma wrote:
bmahes wrote:
IanStewart wrote:
The conventional approach here would be to find the equation of the line. You can find the slope by subtracting the y coordinates of the two points, and then dividing by the difference in x coordinates:

m = (0 - 5)/(7 - 0) = -5/7

The y intercept of the line is 5, since (0,5) is on the line, so the equation of the line is

y = (-5/7)x + 5

If a point is on this line, then the equation above must be true if we plug in that point's coordinates. So we can now plug in each answer choice to see which works. If you plug in answer D (x = 14, y = -5), you get:

-5 = (-5/7)(14) + 5
-5 = -10 + 5
-5 = -5

so the equation is true, and D is on the line.

If you have a good conceptual understanding of slopes, you can bypass the calculations above. If a line travels from (0,5) to (7,0), it goes across 7 units and falls 5 units. So if we go across a further 7 units from the point (7,0), we must fall by a further 5 units. We'd then be at point (14,-5).

So I wanted to get this "good conceptual understanding of slopes" and not solve it by making equations. Can you please help me with it?



I have discussed the concept of slope here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2016/0 ... line-gmat/

The points on the line are (0, 5) and (7, 0). So for 7 units increase in x value, y decreases by 5 units.

If x reduces by 7 units, y will increase by 5 units. So from (0, 5) we will get that (-7, 10) will lie on the line, not (-7, 5)
If x increases by 14 units, y will reduce by 10 units so from (0, 5), we will get (14, -5). This is option (D). Correct answer.
Some more examples:
If x reduces by 14 units, y will increase by 10 units so from (0, 5), we will get (-14, 15), not (-14, 10)
If x increases by 21 units, y will reduce by 15 units so from (0, 5), we will get (21, -10), not (21, -9)


Hi VeritasKarishma

I have read your blog (highly helpful) and have understood that if the slope here were +2 it would mean if x increases by 1 unit, y would increase by 2 units. In this question, the slope is -5/7 (a fraction) now how do we apply this logic here? How did you come to this conclusion "for 7 units increase in x value, y decreases by 5 units"
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   05 May 2020, 09:28
A line passes through infinite number of points...so if we take any pair of points it should give the same slope.
Let us call the given points A(0,5) & B(7,0)
So slope of AB is -5/7
Now the answer we choose if it lies on the same line, that is if the line passes through a point, that point with either point A or point B should give the same slope.
Let us look at choice A (-14,10) with (0,5) gives a slope of -5/14
B (-7,5) with (0,5) gives 0 slope
C (12,-4) with (0,5) gives slope of -3/4
D choice (14,-5) with (0,5) gives the required slope of -5/7
Hence D

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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   11 May 2020, 14:40
(0,5) and (7,0) are x and y intercept form. So we can write the equation of a line in the intercept form. x/a + y/b = 1. So the equation become x/7 + y/5 = 1. Putting the values of x and y from the answer choice we can find the option which satisfies the equation. Ans D.
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   15 Jan 2021, 10:29
IanStewart wrote:
If you have a good conceptual understanding of slopes, you can bypass the calculations above. If a line travels from (0,5) to (7,0), it goes across 7 units and falls 5 units. So if we go across a further 7 units from the point (7,0), we must fall by a further 5 units. We'd then be at point (14,-5).


Very nice explanation! Thank you Ian.
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Re: In the rectangular coordinate system, a line passes through the points [#permalink] New post   26 May 2023, 05:50
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