Math Is Fun Forum

Rearrange into classic y=mx+c:

y=2x+5
y=2x+10

Hey, they have the same slope (parallel) !

The line we are trying to find has the equation y=mx+c, but c is 0 because you said it passes through (0,0) so it is really just:

y=mx (where m is the slope we want to find)

This should intersect the first line at 2x' + 5 = mx'  ==> (2-m)x' + 5 = 0  ==> x'= 5 / (m-2)
... and the y value is y'=2x'+5 = 2 (5 / (m-2)) + 5 = 10/(m-2) + 5

and should intersect the second line at 2x'' + 10 = mx''  ==> (2-m)x'' + 10 = 0  ==> x''= 10 / (m-2)
... and the y value is y'=2x'+10 = 2 (10 / (m-2)) + 10 = 20/(m-2) + 10

Because we have a LOT of 1/(m-2) terms lets call that "k"

So, the intersection points are:
1) x'=5k and y'=10k+5
2) x''=10k and y''=20k+10

The distance between these two points is

d = sqrt( (x''-x')^2 + (y''-y') ) which you say is also equal to sqrt(10)

sqrt(10) = sqrt( (x''-x')^2 + (y''-y') ) = sqrt( (10k-5k)^2 + (20k+10 - 10k -5)^2 ) = sqrt ( (5k)^2 + (10k+5)^2 )

10 = (5k)^2 + (10k+5)^2 = 25 k^2 + 25 (2k+1)^2 = 25 k^2 + 25 (4k^2 + 4k + 1)

Subtract 10 and gather terms: 0 = 25 (1+4) k^2 + 100 k + 25 - 10

Neaten Up: 125 k^2 + 100 k + 15 = 0

Divide by 5: 25 k^2 + 20 k + 3 = 0

Factor: (5k+3)(5k+1) = 0

The roots are: k = -(3/5) and -(1/5)  (if you don't believe me just try them)

... but we didn't want k we wanted m ... !!

k=1/(m-2), so we solve that knowing k and get: m = 1/3 and m = -3

So there are two slopes that work!

THE TWO SOLUTIONS ARE:

y = (1/3)x

y = -3x

NOTE 1: because this is a long solution, we skipped over some minor equation solving (nothing too hard!)

NOTE 2: there may be a simpler geometrical solution to this!

(Solution not by me, but by "Astronomer")

Hi, chantelle.

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Thank you !

But "All" was a pretty weak character anyway - too stereotyped and no interesting character flaws. Just my opinion, mind.

Yes, indeed ! As the hillbilly said to the teacher "Mah boy needs as mahch trigger-nometry as he can git - he cain't shoot straight 'tall!"

I have a page which describes converting from bearing/distance (known as Polar Coordinates) to East and North (known as Cartesian Coordinates) here

I don't have a page which describes adding two bearings/distances yet. But the steps are: 1) convert to cartesian, 2) add the x's and y's, 3) convert back to polar

Indeed!

When books are placed on a shelf, the pages are like:

[        Volume 1        ][        Volume 2         ][        Volume 3        ]
[Page 1000 ... Page 1][Page 1000 ... Page 1][Page 1000 ... Page 1]

So, it is a tricky puzzle. I could have made it harder by saying the worm chews 100 SHEETS in an hour and you may have fallen into the trap of sheets vs pages, but I didn't (because I am nice )

Yep, just recently covered here.

The question was adding up the numbers 1,2,3,4,5 (your problem is just like 5,4,3,2,1) but my answer included being able to add things like 5,6,7,8,9 (ie the general solution) ... and Milos rounded out the whole discussion.

I have added your solution here.

Start: sqrt(11 - x)   -   sqrt(x + 6)  = 3

Move sqrt(x + 6) to other side: sqrt(11 - x) = 3 + sqrt(x + 6)

Square both sides: [sqrt(11 - x)]^2 = (3 + sqrt(x + 6))^2  =>  11 - x = (3 + sqrt(x + 6))^2

Expand RHS: 11 - x = 3^2 + 2*3*sqrt(x + 6) + sqrt(x + 6)^2

Simplify: 11 - x = 9 + 6 * sqrt(x + 6) + (x+6)

More: 11 - x = x + 15 + 6 * sqrt(x + 6)

More: 11 - x - x - 15 = 6 * sqrt(x + 6)

More: -4 - 2x = 6 * sqrt(x + 6)

Square both sides: (-4 - 2x)^2 = (6 * sqrt(x + 6))^2

Expand: (-4)^2 + -4*-2x + -2x*-4 + (-2x)^2 = 36 * (x+6)

Simplify: 16 + 2(8x) + 4x^2 = 36(x+6)

More: 16 +16x + 4x^2 = 36x + 216

More: 16-216 + 16x-36x + 4x^2 = 0

More: -200 -20x +4x^2 = 0

Divide by 4: x^2 - 5x - 50 = 0

Its a nice quadratic now!

Good first step!

Then (adding 1 and 1 to make 2 ): 1 + 1/ [ (2-x)/(1-x) ]

Inverting the fraction: 1 + (1-x)/(2-x)

Common Denom: (2-x)/(2-x) + (1-x)/(2-x)  => [ (2-x) + (1-x) ] / (2-x)

Simplify Top: (3-2x) / (2-x)

That is as far as I can go in simplifying the term.

... and they can get at you from unexpected places !

Good maths, but look at the books on a shelf.