Solution of the equation 4/x-5/y=x+y/xy+3/10 and 3xy=10(y-x)
Solve for x and y : 4/x-5/x=x+y/xy +3/10 and 3xy =10(y-x)
how about some spaces or parentheses to show factors?
Let's solve the given equations step-by-step.
Equation 1: 4/x - 5/y = x + y/xy + 3/10
To simplify the equation, we can multiply both sides by 10xy to eliminate the denominators:
10xy * (4/x - 5/y) = 10xy * (x + y/xy + 3/10)
Simplifying further:
40y - 50x = 10xy^2 + 10x^2 + 3xy
Rearranging the terms:
10xy^2 + 10x^2 + 3xy - 50x - 40y = 0
Now, let's solve Equation 2: 3xy = 10(y - x)
Expanding:
3xy = 10y - 10x
Rearranging the terms:
3xy + 10x - 10y = 0
We now have two equations:
1) 10xy^2 + 10x^2 + 3xy - 50x - 40y = 0
2) 3xy + 10x - 10y = 0
To proceed, we can use a method called substitution:
From Equation 2, we can rearrange it to solve for x:
3xy + 10x - 10y = 0
10x = 10y - 3xy
x = (10y - 3xy)/10
Now, substitute this value of x in Equation 1:
10xy^2 + 10x^2 + 3xy - 50x - 40y = 0
10y^3 - 3y^2x + 10(10y - 3xy)^2 + 3xy - 50(10y - 3xy) - 40y = 0
Expand and simplify the equation further:
10y^3 - 3y^2(10y - 3xy) + 10(100y^2 - 60xy + 9x^2y) + 3xy - 500y + 150xy - 40y = 0
10y^3 - 30y^3 + 9xy^2 + 1000y^2 - 600xy + 90x^2y + 3xy - 500y + 150xy - 40y = 0
-20y^3 + 1000y^2 - 20y - 500y + 3xy + 9xy^2 + 90x^2y + 150xy = 0
-20y^3 + 1000y^2 - 520y + 192xy + 90x^2y = 0
Factor out y:
-20y(y^2 - 50y + 26 - 9xy) + 90xy(2y + x) = 0
Now we have two possible solutions:
1) y = 0
By substituting y = 0 in Equation 2, we get x = 0.
2) y^2 - 50y + 26 - 9xy = 0 ---(Equation 3)
Now, we can use Equation 3 to find the value of x in terms of y. Note that there is no simple way to solve it analytically for x. It can be solved numerically using methods such as iterative methods or graphical methods.
So, the possible solutions are (x, y) = (0, 0) and the solution(s) obtained from solving Equation 3 numerically.
To solve the given equation, let's start by simplifying it step by step:
Equation 1: 4/x - 5/y = x + y/(xy) + 3/10
Equation 2: 3xy = 10(y-x)
Step 1: Eliminate fractions in Equation 1.
Multiply the entire equation by x*y*10 to get rid of the denominators:
10*4*y - 10*5*x = 10*x*y + 10*y + x*y*3
40y - 50x = 10xy + 10y + 3xy
Step 2: Combine like terms.
Rearrange the equation to group like terms:
(10xy + 3xy) + (40y - 10y) = (-50x)
13xy + 30y = -50x
Step 3: Substitute Equation 2 into Equation 1.
Replace 10(y-x) in Equation 1 with 3xy:
13xy + 30y = -50x
Step 4: Rearrange the equation.
Move all terms to one side of the equation:
13xy + 30y + 50x = 0
Step 5: Simplify the equation.
Divide the entire equation by 10 to simplify it:
(13/10)xy + 3y + 5x = 0
Now, we have two equations:
Equation 1: 13xy + 30y + 50x = 0
Equation 2: 3xy = 10(y-x)
To solve these equations simultaneously, we can use substitution or elimination method. Let's solve it using the substitution method.
From Equation 2, we can solve for x:
3xy = 10y - 10x
10x = 10y - 3xy
x = (10y - 3xy)/10
x = y - (3/10)xy
Substituting this value of x into Equation 1, we have:
13( y - (3/10)xy )y + 30y + 50( y - (3/10)xy) = 0
Now, simplify and solve for y:
13y^2 - (39/10)xy^2 + 30y + 50y - (15/10)xy = 0
13y^2 + (78/10)xy - (13/10)xy^2 + 80y = 0
130y^2 + 780xy - 13xy^2 + 800y = 0
130y^2 + 800y + 780xy - 13xy^2 = 0
This is a quadratic equation in terms of y. We can solve it by factoring, completing the square, or using the quadratic formula. The specific method would depend on the given values of x and y or any additional information provided.