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 + x y + x = 81
x ( x + y + 1 ) = 81 Divide both sides by x
x + y + 1 = 81 / x
y ^ 2 + x y + y = 51
y ( y + x + 1 ) = 51
y ( x + y + 1 ) = 51 Divide both sides by y
x + y + 1 = 51 / y
x + y + 1 = x + y + 1
81 / x = 51 / y Multiply both sides by x y
81 x y / x = 51 x y / y
81 y = 51 x Divide both sides by 81
81 y / 81 = 51 x / 81
y = 51 x / 81
y = 3 * 17 * x / ( 3 * 27 )
y = 17 x / 27
Now put this value in formula :
x ( x + y + 1 ) = 81
x ( x + 17 x / 27 + 1 ) = 81
x ( 27 x / 27 + 17 x / 27 + 1 ) = 81
x ( 44 x / 27 + 1 ) = 81
44 x ^ 2 / 27 + x = 81
44 x ^ 2 / 27 + x - 81 = 0 Multiply both sides by 27
44 x ^ 2 + 27 x - 2187 = 0
The exact solutions of this equation are :
x = - 81 / 11 and x = 27 / 4
For x = - 81 / 11
y = 17 x / 27 = - 51 / 11
x + y = - 81 / 11 - 51 / 11 = - 132 / 11 = - 12
For x = 27 / 4
y = 17 x / 27 = 17 / 4
x + y = 27 / 4 - 17 / 4 = 44 / 4 = 11
x + y is positive so :
x = 27 / 4 , y = 17 / 4
x + y = 11
For x = 27 / 4
y = 17 x / 27 = 17 / 4
x + y = 27 / 4 + 17 / 4 = 44 / 4 = 11
To determine the value of x+y, we need to find the values of x and y that satisfy both conics. Let's solve this step by step:
1. Start by rearranging the first equation, x^2 + xy + x = 81, to isolate x:
x^2 + xy + x - 81 = 0
2. Similarly, rearrange the second equation, y^2 + xy + y = 51, to isolate y:
y^2 + xy + y - 51 = 0
3. Now, since (x, y) lies on both conics, we can combine the equations by setting them equal to each other:
x^2 + xy + x - 81 = y^2 + xy + y - 51
4. Simplify the equation by canceling out similar terms:
x^2 + xy + x - y^2 - xy - y + 30 = 0
5. Rearrange the equation to isolate x and y terms separately:
(x^2 - y^2) + (xy - xy) + (x - y) + 30 = 0
Notice that the xy terms cancel out.
6. Factor the difference of squares (x^2 - y^2) to simplify the equation further:
(x - y)(x + y) + (x - y) + 30 = 0
7. Combine like terms:
(x - y)(x + y + 1) = -30
8. Now, there are multiple possible values of x and y that could satisfy this equation. However, we know that x + y should be positive.
9. We need to determine the possible factors of -30 that lead to positive values of x + y. Let's list the factors:
-30 = -1 * 30 = -2 * 15 = -3 * 10 = -5 * 6
10. From these factors, we can see that the only combination where x + y is positive is when x - y = -6 and x + y + 1 = 5.
11. Solving the system of equations:
From x - y = -6, we can add this equation to x + y + 1 = 5 to get:
2x + 1 = -1
2x = -2
x = -1
Substituting the value of x into x + y + 1 = 5:
-1 + y + 1 = 5
y = 5 - 1
y = 4
12. Therefore, the values of x and y that satisfy both conics and make x + y positive are x = -1 and y = 4.
13. Finally, we can calculate the value of x + y:
x + y = -1 + 4 = 3
So, the value of x + y is 3.