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In the rectangular coordinate system, a line passes through the points  [#permalink]

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Question Stats: 70% (02:12) correct 30% (01:55) wrong based on 267 sessions

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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]

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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]

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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]

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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]

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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]

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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 .
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Re: In the rectangular coordinate system, a line passes through the points  [#permalink]

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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.veritasprep.com/blog/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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