The point (x,y) lies on both conics x2+xy+x=81 and y2+xy+y=51. Given that x+y is positive, determine the value of x+y.
x^2 + xy + x = 81
xy + y^2 + y = 51
add them to get
x^2 + 2xy + y^2 + x + y = 132
(x+y)^2 + (x+y) - 132 = 0
((x+y)+13)((x+y)-12) = 0
so, x+y = 12
To find the value of x+y, we need to solve the given system of equations x^2+xy+x=81 and y^2+xy+y=51 simultaneously.
Let's start by subtracting 81 from both sides of the first equation:
x^2 + xy + x - 81 = 0
Similarly, subtract 51 from both sides of the second equation:
y^2 + xy + y - 51 = 0
Now we have a system of equations:
x^2 + xy + x - 81 = 0
y^2 + xy + y - 51 = 0
To solve this system, we can use the method of substitution.
Let's solve the first equation for y:
xy + y = 81 - x^2 - x
y(x + 1) = (81 - x)(81 + x)
y = (81 - x)(81 + x) / (x + 1)
Substitute this expression for y into the second equation:
((81 - x)(81 + x) / (x + 1))^2 + x((81 - x)(81 + x) / (x + 1)) + (81 - x)(81 + x) / (x + 1) - 51 = 0
Now, we can simplify the equation and solve for x:
((81 - x)(81 + x))^2 + x((81 - x)(81 + x)) + (81 - x)(81 + x) - 51(x + 1) = 0
After simplifying the equation, we get:
(81 - x)^3 + (81 - x)(81 + x)^2 - 51(x + 1)(x + 1) = 0
We can then solve this equation using algebraic methods such as factoring, expanding, or simplifying. Once we determine the value of x, we can substitute it back into any of the original equations to find the corresponding value of y. Finally, x+y will give us the desired value.