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Co-ordinate planes

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by vinay1983 » Thu Sep 19, 2013 5:44 am
ln the coordinate plane, line k passes through the origin and has slope 2. lf points (3,y) and (x,4) are on line k, then x+y=?

(A) 3.5
(B) 7
(C) B
(D) 10
(E) 14

I understood that we have to substitute values of x and y from the given co-ordinate points to get the unknown co-ordinate points. My question is "The OG explanation assumes equation of line of k to be y=2x". How can we know this?Is there any concept hidden?
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by Brent@GMATPrepNow » Thu Sep 19, 2013 5:58 am
vinay1983 wrote:ln the coordinate plane, line k passes through the origin and has slope 2. lf points (3,y) and (x,4) are on line k, then x+y=?

(A) 3.5
(B) 7
(C) 8
(D) 10
(E) 14

I understood that we have to substitute values of x and y from the given co-ordinate points to get the unknown co-ordinate points. My question is "The OG explanation assumes equation of line of k to be y=2x". How can we know this?Is there any concept hidden?
It often helps to write equations for lines in the form y = mx + b (this is called slope y-intercept form), where m = the slope of the line, and b = the y-intercept of the line.

The question tells us that the line has slope 2. So, m = 2
The question also tells us that the line passes through the origin (0,0). So, the y-intercept is 0, which means b = 0
So, the equation of the line is y = 2x + 0, or just y = 2x

If the point (3,y) is on the line, then its coordinates must satisfy the equation y = 2x
So, plug x=3 and y=y into the equation to get y = (2)(3) = 6

If the point (x,4) is on the line, then its coordinates must satisfy the equation y = 2x
So, plug x=x and y=4 into the equation to get 4 = 2x, which means x = 2

So, x + y = 2 + 6 = [spoiler]8 = C[/spoiler]

Cheers,
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by [email protected] » Thu Sep 19, 2013 12:36 pm
Hi vinay1983,

Brent has provided the full math explanation of the concept; the "rule" that you're looking for is this:

By passing through the origin, the line has a y-intercept of 0, so the equation of the line will almost always be y = (slope)(x)

The exception would be the line x = 0, which would be a line that goes straight up-and-down and is ON the y-axis.

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by Stuti567 » Mon May 02, 2016 11:28 pm
Hi,

I understood the explanation, however, I tried a different approach to get the answer but I'm not getting it!

I used the slope formula to find the equation of the line. i.e ([4-y)/(x-3)] = 2 which gives 2x + y = 10 which cannot satisfy the origin. I cannot understand the reason for this discrepancy. Please explain.

Thanks.
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by 800_or_bust » Tue May 03, 2016 3:09 am
vinay1983 wrote:ln the coordinate plane, line k passes through the origin and has slope 2. lf points (3,y) and (x,4) are on line k, then x+y=?

(A) 3.5
(B) 7
(C) B
(D) 10
(E) 14

I understood that we have to substitute values of x and y from the given co-ordinate points to get the unknown co-ordinate points. My question is "The OG explanation assumes equation of line of k to be y=2x". How can we know this?Is there any concept hidden?
A line with a slope of 2 that passes through the origin must be represented by the equation y = 2x. Remember "slope y-intercept" form. The slope is 2, and the y-intercept is 0.
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by GMATGuruNY » Tue May 03, 2016 4:51 am
Stuti567 wrote:Hi,

I understood the explanation, however, I tried a different approach to get the answer but I'm not getting it!

I used the slope formula to find the equation of the line. i.e ([4-y)/(x-3)] = 2 which gives 2x + y = 10 which cannot satisfy the origin. I cannot understand the reason for this discrepancy. Please explain.

Thanks.
The general equation form for a line is as follows:
ax + by = c.
You are attempting to represent line k in this form.
In this equation, x and y represent a POINT ON THE LINE.

The prompt above:
Points (3, y) and (x ,4) are on line k.
Here, x and y do NOT represent a point on the line, since these values belong to TWO DIFFERENT POINTS.

Thus, we cannot generate ax + by = c from (3, y) and (x, 4).

Your approach would be valid for the following prompt:
ln the coordinate plane, line k passes through the origin and has slope 2. lf points (3,6) and (x,y) are on line k, then x+y=?
Here, since (x, y) is a point on the line and the slope = 2, we can proceed as follows:
(y-6)/(x-3) = 2
y-6 = 2x-6
y=2x.
The result is an equation for a line that passes through the origin.
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by harmeen19 » Sat Oct 08, 2016 1:47 am
A different approach.
As per the slope(m) formula for a line having two points (x,y) & (X,Y) => (Y-y)/(X-x)= m
We have 2 points i.e., (3,y) & (x,4)
So the equation becomes, (4-y)/(x-3)=2, since slope is given as 2.

Therefore, 2x-6=4-y
Simplifying the above, 2x=10-y ---eq1

Now, since we are told the line passes through origin, so the point (0,0) should also lie on the line.
With this, we can take the point (0,0) and any of the two given points and substitute them into the slope formula to find either x or y.
I.e., if we take (0,0) & (3,y) then, (0-y)/(0-3)=2 => y=6
And hence, from eq1, x=2.

So, x+y=8

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by Scott@TargetTestPrep » Fri Dec 08, 2017 10:53 am
vinay1983 wrote:ln the coordinate plane, line k passes through the origin and has slope 2. lf points (3,y) and (x,4) are on line k, then x+y=?

(A) 3.5
(B) 7
(C) 8
(D) 10
(E) 14
We are given that line k passes through the origin (or the point (0,0)) and has a slope of 2. Since slope = (change in y)/(change in x), we can create the following equation using the coordinates (0,0) and (3,y):

2 = (y - 0)/(3 - 0)

2 = y/3

y = 6

We can use the slope equation again, this time using the coordinates (0,0) and (x,4):

2 = (4 - 0)/(x - 0)

2 = 4/x

2x = 4

x = 2

Thus, x + y = 6 + 2 = 8.

Answer: C

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