For every point (a, b) on Line 1, there is a corresponding point

Question

For every point  (a, b)  on Line 1, there is a corresponding point  (b, -a)  on Line 2. Given that the equation of Line 1 is  y = 2x + 1 , what is the equation of Line 2?

A.  y = \frac{x}{2} + \frac{1}{2}  
B.  y = -\frac{x}{2} + \frac{1}{2} 
C.  y = -\frac{x}{2} - \frac{1}{2} 
D.  y = \frac{x}{2} - \frac{1}{2} 
E.  y = -\frac{x}{2} +1

Bunuel · Accepted Answer

For every point  (a, b)  on Line 1, there is a corresponding point  (b, -a)  on Line 2. Given that the equation of Line 1 is  y = 2x + 1 , what is the equation of Line 2?

A.  y = \frac{x}{2} + \frac{1}{2}  
B.  y = -\frac{x}{2} + \frac{1}{2} 
C.  y = -\frac{x}{2} - \frac{1}{2} 
D.  y = \frac{x}{2} - \frac{1}{2} 
E.  y = -\frac{x}{2} +1

To find the equation of Line 2, we can first identify two points on Line 1 and then determine their corresponding points on Line 2. Points  (0, 1)  and  (1, 3)  lie on Line 1, so their corresponding points on Line 2 are  (1, 0)  and  (3, -1) , respectively.

Using the form  y = mx + b  for the equation of Line 2, we can employ the two points we found to create a system of equations to solve for the slope  m  and the y-intercept  b . Plugging the coordinates of the points into the equation, we obtain:

0 = m(1) + b

-1 = m(3) + b

Upon solving this system of equations, we find that  m = -\frac{1}{2}  and  b = \frac{1}{2} .

Consequently, the equation of Line 2 is  y = -\frac{x}{2} + \frac{1}{2} .

Answer: B

Mackieman · Answer

The formula for finding the slope of a line is (change in y)/(change in x)

In your case, two coordinates (1,0) and (0,1/2) where (x,y)

(0-1/2)/(1-0) = -1/2 which is the slope of the line. The common formula for a line is y = kx + m where k is the slope and m is the y-intercept.

The second coordinate (0,1/2) tells us that when x = 0 y = 1/2, which is the y-intercept or 'm' in the formula y = kx+m where m is the y-intercept and 'k' is the slope. Plugging in the slope (-1/2) and the intercept (1/2) into the formula gives us y = 1/2-(x/2)

multiplying by 2 on both sides:
2y = 1-x

tinki · Answer

thanks . yet im actually wondering what is the formula used here that comes to direct equation. ???

The equation of line 2 is \frac{y - 0}{\frac{1}{2} - 0} = \frac{x - 1}{0 - 1} or 2y = 1 - x

tinki · Answer

sorry, here is the equation of my interest :

The equation of line 2 is  \frac{y - 0}{\frac{1}{2} - 0} = \frac{x - 1}{0 - 1}  or  2y = 1 - x  .

tinki · Answer

y- intercept is at x=0 as i know. so y= 1/2 when x=0 Am i missing something?

Mackieman · Answer

Sorry, you are correct, I had a rough day yesterday ;-)

I hope it makes sense now. (edited my post above)

Bunuel · Answer

Check this: math-coordinate-geometry-87652.html (chapter "Lines in Coordinate Geometry").

tinki · Answer

I saw the link. VERY IMPRESSIVE!!!! + kudo from me

GREAT JOB !!!

Sachin9 · Answer

bunuel/Karishma

This is how I solved and it worked but I don't know why it worked :D Please enligthen me about why it worked...

I plugged in (a,b) in line 1 equeation to get b=2a+1..

then I started plugging in (b,-a) in the answer choices.. the 2nd answer choice resulted in b=2a+1.. and hence I marked B..
I am not sure why this worked.. Please help. .

KarishmaB · Answer

Responding to a pm:

a and b stand for two numbers which define a co-ordinate on the plane. When you say that (a, b) lies on y = 2x+1, it means the relation between a and b is b = 2a + 1. e.g. if a = 0, b = 1; if a = 1, b = 3... At the end of the day, a line is nothing but a depiction of how one variable changes with another. A line just shows you the relation between 2 variables.

If (b, -a) lies on a line 2y = 1-x, this is just a different way of expressing the same relation between the two numbers a and b. 
a and b are the same set of numbers (i.e. if a = 0, b = 1; if a = 1, b = 3...)
So after manipulating the equation a little, you are bound to get b = 2a + 1 only.

As you figured out, the approach is a little un-intuitive. When I looked at the problem, I actually solved it exactly the same way except that I took numbers rather than a and b.

I said, if (a, b) lies on y = 2x + 1, if a = 1, b = 3.
So (3, -1) must lie on the new equation of the line. When I put (3, -1) in the options, I see that only (B) satisfies.

legitpro · Answer

it took me a while to get this

so what i did was create a value for x then sub it into line 1 to find out what y is. now that you have (a,b) create the point (b,-a).

now what you do is sub is points of line 2 into various equations until you find one that makes sense.

AnuK2222 · Answer

Correct answer B, GMAT club questions give me a reality check like nothing else

I found points where line one intercept x and y axis i.e (-1/2,0) and (0,1) as (a,b) corresponds to (-b,a)..line 2 meets x and y axis at (1,0) and (0,1/2) using 2 points we find slope = 0-(1/2)/1-0 = -1/2 we can see y intercept is 1/2 (point where line 2 intersects y axis) line 2=> y=(-1/2)x + 1/2 hope it helps

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For every point (a, b) on Line 1, there is a corresponding point

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