$$\frac{10x}{5x-9}+\frac{14x}{2x+9}=9$$
$x = \frac{81}{31} = 2\frac{19}{31} \approx 2.612903226$
$$\left(2x+9\right)\times 10x+\left(5x-9\right)\times 14x=9\left(5x-9\right)\left(2x+9\right)$$
$$\left(20x+90\right)x+\left(5x-9\right)\times 14x=9\left(5x-9\right)\left(2x+9\right)$$
$$20x^{2}+90x+\left(5x-9\right)\times 14x=9\left(5x-9\right)\left(2x+9\right)$$
$$20x^{2}+90x+\left(70x-126\right)x=9\left(5x-9\right)\left(2x+9\right)$$
$$20x^{2}+90x+70x^{2}-126x=9\left(5x-9\right)\left(2x+9\right)$$
$$90x^{2}+90x-126x=9\left(5x-9\right)\left(2x+9\right)$$
$$90x^{2}-36x=9\left(5x-9\right)\left(2x+9\right)$$
$$90x^{2}-36x=\left(45x-81\right)\left(2x+9\right)$$
$$90x^{2}-36x=90x^{2}+243x-729$$
$$90x^{2}-36x-90x^{2}=243x-729$$
$$-36x=243x-729$$
$$-36x-243x=-729$$
$$-279x=-729$$
$$x=\frac{-729}{-279}$$
$$x=\frac{81}{31}$$
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