We're given the following restrictions: $$x < y$$
$$-29x^3+15y^3=-61209$$
and
$$-60x-90y=-3420$$
Using these three things, we need to find the value of:
$$-50x^2+70y^2$$
Is there a simple approach to solve this? Solving both equations for $x$ and $y$ and plugging in the required result, makes things very complicated and I'm hoping that there is an easier way of doing this.
We need to solve the following system. $$29x^3-15y^3=61209$$ and $$2x+3y=114,$$ which gives $$29x^3-15y^3=\frac{61209(2x+3y)^3}{114^3}$$ or $$54872(29x^3-15y^3)=2267(2x+3y)^3$$ or $$524384x^3-27204x^2y-40806xy^2-294763y^3=0$$ or $$524384x^3-458836x^2y+431632x^2y-377678xy^2+336872xy^2-294763y^3=0$$ or $$65548x^2(8x-7y)+53954xy(8x-7y)+42109y^2(8x-7y)=0$$ or $$(8x-7y)(65548x^2+53954xy+42109y^2)=0,$$ which gives $$8x-7y=0$$ and the rest is smooth.
We can get a factor $8x-7y$ by the following way. $$524384=2^5\cdot7\cdot2341$$ and $$294763=7\cdot17\cdot2477.$$ Now, if there is a factor $ax+by$, where $a$ and $b$ are integers,
so $a$ divides $2^5\cdot7\cdot2341$ and $b$ divides $7\cdot17\cdot2477.$
We see that $a=8$ and $b=-7$ are valid.