How to solve this system?
x + [(3x-y)/(x^2+y^2)] = 3
y - [(x+3y)/(x^2+y^2)] = 0
To solve this system of equations, we can use substitution method or elimination method. Let's use the substitution method in this case.
Step 1: Start by isolating one variable in one of the equations. Let's isolate y in the second equation:
y - [(x+3y)/(x^2+y^2)] = 0
Move [(x+3y)/(x^2+y^2)] to the other side:
y = [(x+3y)/(x^2+y^2)]
Step 2: Substitute the value of y from the second equation into the first equation:
x + [(3x-y)/(x^2+y^2)] = 3
Replace y with [(x+3y)/(x^2+y^2)]:
x + [(3x-[(x+3y)/(x^2+y^2)])/(x^2+y^2)] = 3
Step 3: Simplify the equation by getting rid of the denominators:
Multiply both sides of the equation by (x^2+y^2):
(x^2+y^2)(x) + (3x-[(x+3y)/(x^2+y^2)])(x^2+y^2) = 3(x^2+y^2)
Simplify each part of the equation:
x^3 + xy^2 + 3x(x^2+y^2) - (x+3y) = 3x^2 + 3y^2
Simplify further:
x^3 + xy^2 + 3x^3 + 3xy^2 - x - 3y = 3x^2 + 3y^2
Combine like terms:
4x^3 + 6xy^2 - x - 3y = 3x^2 + 3y^2
Step 4: Rearrange the equation to have all terms on one side:
4x^3 + 6xy^2 - x - 3y - 3x^2 - 3y^2 = 0
Simplify further:
4x^3 + 6xy^2 - x - 3x^2 - 3y - 3y^2 = 0
Step 5: Now we have one equation with only one variable (x). We can solve this equation using various methods like factoring, graphing, or numerical methods. Once we find the value of x, we can substitute it back into one of the original equations to solve for y.