Rational Expression

Hello all,

as for this problem,

given that 9 / y - 4 / x = 5 / (x+y)

9x - 4y / xy = 5 / (x+y)

(x+y)(9x - 4y) = 5(xy)

9x^(2) - 4xy + 9xy - 4y^(2) = 5xy

9x^(2) - 4y^(2) = 0

9x^(2) = 4y^(2)

(3x)^(2) = (2y)^(2)

by taking square root on both sides,

3x = 2y

3x / 2y = 1

therefore, | 3x / 2y | =1......

thanks....

Multiply everything by $(x)(y)(x+y)$ to get rid of the denominators to get $9x(x+y)-4y(x+y)=5xy$ . Distribute and expand to get $9x^2+5xy-4y^2=5xy$ , then simplify to get $9x^2=4y^2$ . Take the square root and isolate a variable to get $x=\pm\frac{2}{3}y$ . Plugging in this to the desired value of $|\frac{3x}{2y}|$ shows the answer is $\boxed{1}$ .

Desenvolvendo,

$\frac{9}{y} - \frac{4}{x} = \frac{5}{x + y} \\\\ \frac{9x - 4y}{xy} = \frac{5}{x + y} \\\\ 9x^2 + 9xy - 4xy - 4y^2 = 5xy \\ 9x^2 = 4y^2 \\\\ \frac{x^2}{y^2} = \frac{4}{9} \\\\ |\frac{x}{y}| = \frac{2}{3}$

Com efeito,

$|\frac{3x}{2y}| = \\\\ \frac{3}{2} \cdot |\frac{x}{y}| = \\\\ \frac{3}{2} \cdot \frac{2}{3} = \\\\ \boxed{1}$

$9 \div y - 4 \div x = 5 \div ( x + y ) \rightarrow 9x \div - 4 = 5x \div y \rightarrow 9x - 4y = 5xy \rightarrow x = 1, y = 1$ .

So $\boxed { 1 }$ .