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For every point \((a, b)\) lying on line 1, point \((b, -a)\) lies on line 2. If the equation of line 1 is \(y = 2x + 1\) , what is the equation of line 2 ?

Find two points on line 2 and use their coordinates to build the line's equation. Points \((0, 1)\) and \((-\frac{1}{2}, 0)\) on line 1 correspond to points \((1, 0)\) and \((0, \frac{1}{2})\) on line 2. The equation of line 2 is \(\frac{y - 0}{\frac{1}{2} - 0} = \frac{x - 1}{0 - 1}\) or \(2y = 1 - x\) . The correct answer is B.

i cant understand what the formula used here for building equation line 2. Can somebody explain? what is the Formula ? thank you

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

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

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

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

Sorry, you are correct, I had a rough day yesterday I hope it makes sense now. (edited my post above)
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For every point \((a, b)\) lying on line 1, point \((b, -a)\) lies on line 2. If the equation of line 1 is \(y = 2x + 1\) , what is the equation of line 2 ?

Find two points on line 2 and use their coordinates to build the line's equation. Points \((0, 1)\) and \((-\frac{1}{2}, 0)\) on line 1 correspond to points \((1, 0)\) and \((0, \frac{1}{2})\) on line 2. The equation of line 2 is \(\frac{y - 0}{\frac{1}{2} - 0} = \frac{x - 1}{0 - 1}\) or \(2y = 1 - x\) . The correct answer is B.

i cant understand what the formula used here for building equation line 2. Can somebody explain? what is the Formula ? thank you

For every point \((a, b)\) lying on line 1, point \((b, -a)\) lies on line 2. If the equation of line 1 is \(y = 2x + 1\) , what is the equation of line 2 ?

Find two points on line 2 and use their coordinates to build the line's equation. Points \((0, 1)\) and \((-\frac{1}{2}, 0)\) on line 1 correspond to points \((1, 0)\) and \((0, \frac{1}{2})\) on line 2. The equation of line 2 is \(\frac{y - 0}{\frac{1}{2} - 0} = \frac{x - 1}{0 - 1}\) or \(2y = 1 - x\) . The correct answer is B.

i cant understand what the formula used here for building equation line 2. Can somebody explain? what is the Formula ? thank you

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. .
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hope is a good thing, maybe the best of things. And no good thing ever dies.

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. .

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.
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