If these are the right equations:

y/x + (y+5)/(x+5) = 1 + 3/25

y/x - (y-5)/(x-5) = 3/25

then I can’t seem to find a solution.

Can anyone else do it?

Here’s my attempt:

y/x = (y-5)/(x-5) + 3/25

y/x = – (y+5)/(x+5) + 1 + 3/25

Multiply first eqn by 25x(x+5)

and second by 25x(x-5)

25y(x+5) + 25x(y+5) = 25x(x+5) + 3x(x+5)

25y(x-5) + 25x(y-5) = 3x(x-5)

Expand

25yx + 125y + 25yx + 125x = 25x^2 + 125x + 3x^2 + 15x

25yx – 125y + 25yx – 125x = 3x^2 – 15x

Combine like terms in each

50yx + 125y = 28x^2 + 15x

50yx – 125y = 3x^2 + 110x

Solve for y in each

y = (28x^2 + 15x)/(50x + 125)

y = (3x^2 + 110x)/(50x – 125)

Find intersections of:

y = f(x) = (28x^2 + 15x)/(50x + 125)

y = g(x) = (3x^2 + 110x)/(50x – 125)

GeoGebra graph:

http://www.wyzant.com/resources/files/264445/no_solution_fraction_problem
Intersection points from graph:

(8.3898886627, 3.8508654854), (-1.4898886625, 0.7881345142)

Check:

(8.3898886627, 3.8508654854):

3.8508654854/8.3898886627 + (3.8508654854+5)/(8.3898886627+5) =? 1 + 3/25

3.8508654854/8.3898886627 + (8.8508654854)/(13.3898886627) =? 28/25

0.45898886626712 + 0.66101113372628 =? 1.12

1.1199999999934 = 1.12 √

3.8508654854/8.3898886627 - (3.8508654854-5)/(8.3898886627-5) =? 3/25

0.45898886626712 - (-1.1491345146)/(3.3898886627) =? 0.12

0.45898886626712 + 0.33898886628469 =? 0.12

0.79797773255181 ≠ 0.12 X ; point doesn’t work

(-1.4898886625, 0.7881345142):

0.7881345142/(-1.4898886625) + (0.7881345142+5)/(-1.4898886625+5) =? 1.12

-0.52898886610596 + (5.7881345142)/(3.5101113375) =? 1.12

-0.52898886610596 + 1.64898886606898 =? 1.12

1.11999999996302 = 1.12 √

0.7881345142/(-1.4898886625) - (0.7881345142-5)/(-1.4898886625-5) =? 0.12

-0.52898886610596 - (-4.2118654858)/(-6.4898886625) =? 0.12

-0.52898886610596 - (4.2118654858)/(6.4898886625) =? 0.12 X point doesn’t work

So I decided to try again using GeoGebra.

First I solved both equations for y:

y/x + (y+5)/(x+5) = 1 + 3/25

y/x - (y-5)/(x-5) = 3/25

y/x + y/(x+5) + 5/(x+5) = 1 + 3/25

y/x - y/(x-5) + 5/(x-5) = 3/25

y/x + y/(x+5) = - 5/(x+5) + 1 + 3/25

y/x - y/(x-5) = - 5/(x-5) + 3/25

y(1/x + 1/(x+5)) = 1 + 3/25 - 5/(x+5)

y(1/x - 1/(x-5)) = 3/25 - 5/(x-5)

y = f(x) = (1 + 3/25 - 5/(x+5))/(1/x + 1/(x+5))

y = g(x) = (3/25 - 5/(x-5))/(1/x - 1/(x-5))

Then I graphed them with GeoGebra:

http://www.wyzant.com/resources/files/264638/hard_fraction_solution
[Note that you could use your graphing calculator to do the same thing. I use GeoGebra because it can generate png pictures that I can show you.]

The intersections of the two graphs are:

(-4.166666666679832, -5.083333333327673), (0,0), and (25,13).

The second can be discarded because we can’t have a zero denominator.

From the third we get the fraction 13/25 as a possible solution.

The first needs to be converted from repeating decimals to fractions:

10 n = 41.66_

-n = -4.16_

9 n = 37.5 = 36 + 3/2

n = 4 + 1/6 = -x

x = –25/6

10 n = 50.833_

-n = -5.083_

9 n = 45.75 = 45 + 3/4

n = 5 + 1/12 = -y

y = -61/25

So the first gives us a possible solution of (-61/25)/(–25/6).

Now let’s check the possible solutions:

13/25 ?

13/25 + 18/30 =

13*6/(6*25) + 18*5/(5*30) =

(13*6 + 18*5)/(5*30) =

(78 + 90)/(5*30) =

168/(5*30) =

2*2*2*3*7/(5*5*2*3) =

2*2*7/(5*5) =

28/25 =

1 + 3/25 √

13/25 - 8/20 =

13*4/(4*25) - 8*5/(5*20) =

(52 - 40)/(5*5*4) =

12/(5*5*4) =

3/25 √

(-61/12)/(-25/6) ?

(-61/12)/(-25/6) + (5-61/12)/(5-25/6) =

61/50 + (60-61)/(60-50) =

61/50 + 1/10 =

61/50 + 5/50 =

56/50 =

28/25 =

1 + 3/25 √

(-61/12)/(-25/6) - (-5-61/12)/(-5-25/6) =

61/50 - (5+61/12)/(5+25/6) =

61/50 - (121/12)/(55/6) =

61/50 - 121/110 =

61/(5*5*2) - 121/(5*2*11) =

61*11/(5*5*2*11) - 121*5/(5*5*2*11) =

(61*11 - 121*5)/(5*5*2*11) =

(671 - 605)/(5*5*2*11) =

66/(5*5*2*11) =

3*2*11/(5*5*2*11) =

3/25 √

=====

So there are two solutions:

13/25 and (-61/12)/(-25/6);

but don’t simplify the second

one before you put it into

the equations. Weird!

Can anyone derive these two solutions analytically?

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