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Can anyone help me to solve this problem:
$x$ and$y$ are real numbers which satisfy $x>y$ and $xy<0$. If
$\left | x \right | + \left | y \right | + \left | 42y-x \right | + \left | 23x-y \right |$
can be expressed as $ax+by$, what is the value of $a+b?$

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since $x>y,xy<0\Longrightarrow x>0,y<0$, so $$42y-x<0, 23x-y>0$$ $$|x|+|y|+|42y-x|+|23x-y|=x-y-(42y-x)+23x-y=x-y-42y+x+23x-y=25x-44y$$ so $$a=25,b=-44,\Longrightarrow a+b=-19$$

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If $x > y$ and $xy < 0$, it must be true that $$y < 0 < x$$


$$|x| + |y| + |23x - y| = x + y + 23x - y = 24x$$

On the other hand, $42y - x < 0$, so $|42 y - x| = x - 42y$.

Can you put it together?

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