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.
mmmh, been messing around this for a while
from 1st:
x(x + y + 1) = 81
from 2nd:
y(y + x + 1) = 51
divide them:
x/y = 81/51
81y = 51x -----> y = 51x/81
sub that into x^2 + xy + x = 81
81x^2 + x(51x/81) + x = 81
81x^2 + 51x^2 + 81x = 6561
132x^2 + 81x - 6561 = 0
x = (-81 ± √3470769)/ 264
= (-81 ± 1863)/264 , which reduces to
= 27/4 or -81/11
ahhh, I guess we could have factored it, lol
132x^2 + 81x - 6561 = 0
44x^2 + 27x - 2187 = 0
(4x - 27)(11x + 81) = 0
anyway, too late,
if x = 27/4 , from y = 51x/81 --> y = 17/4
if x = -81/11, ------> y = -51/11
for first point:
x + y = 27/4 + 17/4 = 44
for 2nd point:
x+y = negative
so for your condition:
x+y = 44
To find the value of x+y, we need to solve the system of equations formed by the given conics.
1. Start by simplifying the equations:
x^2 + xy + x = 81 ... (Equation 1)
y^2 + xy + y = 51 ... (Equation 2)
2. Rearrange Equation 1 to isolate the x terms:
x^2 + xy + x - 81 = 0
3. Similarly, rearrange Equation 2 to isolate the y terms:
y^2 + xy + y - 51 = 0
4. Now, we have a quadratic equation in both x and y:
x^2 + xy + x - 81 = 0 ... (Equation 3)
y^2 + xy + y - 51 = 0 ... (Equation 4)
5. To solve this system of equations, we can use the method of substitution. Solve Equation 3 for x in terms of y:
x = (81 - xy - x)^0.5 ... (Equation 5)
6. Substitute Equation 5 into Equation 4, replacing x with its value from Equation 5:
y^2 + y(81 - xy - x)^0.5 + y - 51 = 0
7. Simplifying the equation, you get:
y^2 + y(81 - y(81 - xy - x)^0.5 - x)^0.5 - 51 = 0
8. Rearrange Equation 7 to isolate the square root term:
y(81 - y(81 - xy - x)^0.5 - x)^0.5 = 51 - y^2
9. Now, we have a quadratic equation in y. You can square both sides of the equation to eliminate the square root term.
10. Upon squaring, we get:
y^2(81 - y(81 - xy - x)^0.5 - x) = (51 - y^2)^2
11. Simplify the equation and collect like terms:
81y^2 - y^3(81 - xy - x)^0.5 - xy^2 - x^2y = (51 - y^2)^2
12. We can now solve for y using numerical methods like substitution or approximation techniques.
13. Once you find the value(s) of y, substitute them back into Equation 5 to find the corresponding values of x.
14. Finally, calculate the sum x+y using the values obtained from the previous step.
Note: Since the equations involve square root terms, the solution may involve complex numbers.