the sum of two natural nos. is 8.determine the nos,.,if the sumof their reciprocals is 8/15?

Question

mention all steps

Answer 1

x + y = 8
1/x + 1/y = 8/15
(y+x)/xy = 8/15
but from the first eqn.
x+y = 8
there fore xy = 15
=> x = 15/y
substituting in the first eqn
15/y + y = 8
taking LCM as y
15+y² = 8y
=>15+y²-8y = 0
is in the form of a quadratic equation which you can solve by using factor theorem
finally the anser turns out to be 3,5
in wich any variable can take up any of these values

Answer 2

the two numbers are x and y.

x + y = 8

The reciprocals are 1/x and 1/y, so

1/x + 1/y = 8/15
get a common denominator:

y/xy + x/xy = 8/15
(y + x)/xy = 8/15

we know x + y = 8, so
8/xy = 8/15
xy = 15

if x + y = 8, then x = 8 - y, so
xy = 15
(8-y)(y) = 15
8y - y^2 = 15
0 = y^2 - 8y + 15
0 = (y - 5)(y - 3)

y = 3 or 5, so x = 5 or 3. The two numbers are 3 and 5.
----------------
You could have also done trial and error
once you knew that xy = 15.

15 = 1 * 15, or 3 * 5

Only 3 * 5 will work since 3 + 5 = 8

Answer 3

Make two linear equations
x + y = 8
1/x + 1/y = 8/15
Flip the second equation
-x - y = 8/15
Therefore, the two equations are:
y = 8-x
-y = 8/15 + x
substitute the -y woth 8-x
-(8 - x) = 8/15 + x
-8 + x = 8/15 + x
-8 + x - x = 8/15
-8 = 8/15
Therefore x = -15
x + y = 8
subsitiute x
-15 + y = 8
Therefore y = 23
X = -15
y = 23

Answer 4

the numbers are 3&5 ok.let 1 no. be x other will be 8-x
1/x+1/8-x=8/15 solution comes x=3,5

Answer 5

Let us assume 1st no.=x
therfore the 2nd no. will be x-8
1/x+1/(8-x)=8/15
8-x+x/x(8-x)=8/15
1/x(8-x)=1/5
x(8-x)=15
8x-x*x=15
x*x+8x-15=0
x*x-8x+15=0
x*x-3x-5x+15
Therofore te no. are 3,5