Solve the simultaneous equations xy= 12 and 1/y - 1/x = 7/12.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to solve for x and y the following system of equations, such that:

`{(xy = 12),(1/y - 1/x = 7/12):}`

You need to start solving the second equation by bringing the terms  to the left side to a common denominator, such that:

`(x - y)/(xy) = 7/12`

Since the first equation indicates that the product `xy = 12` , hence, you may substitute `12`  for `xy`  in the second transformed equation, such that:

`(x - y)/12 = 7/12 `

Reducing duplicate factors yields:

`x - y = 12 => x = 12 + y`

You need to substitute `12 + y`  in the first equation such that:

`(12+y)*y = 12 => 12y + y^2 - 12 = 0`

`y^2 + 12y - 12 = 0 `

Using quadratic formula yields:

`y_(1,2) = (-12+-sqrt(144 + 48))/2 => y_(1,2) = (-12+-sqrt192)/2`

`y_(1,2) = (-12+-8sqrt3)/2 => y_(1,2) = (-6+-4sqrt3)`

`x_(1,2) = 12 - 6+-4sqrt3 => x_(1,2) = 6+-4sqrt3`

Hence, evaluating the solutions to the given system of equations yields `(-6+4sqrt3;6+4sqrt3)`  and `(-6-4sqrt3;6-4sqrt3).`

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misskosmetics | (Level 1) eNoter

Posted on

Thank you for your answers. But the answer I have in my answer keys states the values of x= 8.45 or -1.43 and the values of y= 1.42 or -8.42. How do I get these values?

ayushkhanna's profile pic

ayushkhanna | (Level 1) eNoter

Posted on

then this problem can be taken as follows :

solve 1/y+1/x = 12 

it will come out as : (x+y)/xy = 7/12

put value of xy(given as 12) and take it right side:

x+y=7

now try substituting method, 

only values that give xy=12 are : 2,6 and 3,4.

values 3and 4 also satisfy the newly generated eqauation x+y=7.

 

hence values are 3 and 4.

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