Given the proportion 3x+2y/7=2x+3y/8, determine the value of xy^2-x^2y/x^3+y^3
This is what I got so far...
3x+2y/7=2x+3y/8=k
3x+2y=7k
2x+3y=8k
And that's where I'm stuck at solving this math question.
(3x+2y)/7 = (2x+3y)/8
8(3x+2y) = 7(2x+3y)
24x+16y = 14x+21y
10x = 5y
2x = y
Using that, we have
(xy^2-x^2y)/(x^3+y^3)
= (x(2x)^2 - x^2(2x))/(x^3+(2x)^3)
= (4x^3-2x^3)/(x^3+8x^3)
= 2/9
To solve the system of equations you have obtained, you can use substitution method or elimination method.
Let's use substitution method:
1. Solve one of the equations for one variable (let's solve the second equation for x):
2x + 3y = 8k
2x = 8k - 3y
x = (8k - 3y) / 2
2. Substitute the expression for x in the other equation:
3x + 2y = 7k
3((8k - 3y) / 2) + 2y = 7k
Simplify the equation:
(24k - 9y) / 2 + 2y = 7k
24k - 9y + 4y = 14k
Combine like terms:
24k - 5y = 14k
3. Solve the equation for y:
24k - 5y = 14k
24k - 14k = 5y
10k = 5y
y = 2k
4. Substitute the value of y back into the expression for x:
x = (8k - 3y) / 2
x = (8k - 3(2k)) / 2
x = (8k - 6k) / 2
x = 2k / 2
x = k
Now, we have found the values of x and y in terms of k.
So, x = k and y = 2k.
To determine the value of xy^2 - x^2y / (x^3 + y^3), we substitute the values of x and y into the expression:
xy^2 - x^2y / (x^3 + y^3) =
k(2k)^2 - (k)^2(2k) / (k)^3 + (2k)^3 =
4k^3 - 4k^3 / k^3 + 8k^3 =
0 / 9k^3 =
0
Therefore, the value of xy^2 - x^2y / (x^3 + y^3) is 0.