# Word problem leading to system of linear equations

In his will, a man left his older son 10,000 more than twice as much as he left his younger son. If the estate is worth 497,500 how much did the younger son get?

So far I have.

X - older son

Y - younger son

$497500 = x + y$ I think this one of equations.

$x = 2y$ I think is the other equation.

$497500 = 2y + y$ Solves for younger brother.

$497500 = 3y$

And then I'm confused because I get a fraction for $y$

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In principle, there is nothing wrong with a fractional answer. However, $x=2y+10000$. When you use this, your fraction issue will disappear, and you will get the right answer. – André Nicolas Oct 16 '11 at 19:15
I see how I missed 10000, it all falls together now. – Liger86 Oct 16 '11 at 19:25

If we define $Y$ to be the amount of money left to the younger son and $O$ to be the amount of money left to the older son, then the equation $$Y + O = 497,500$$ says that the entire inheritance is split between them in some way. The equation $$O = 2Y + 10,000$$ captures the condition that the older son received \$10,000 more than twice the amount given to the younger son. Replacing$O$in the first equation gives $$Y + (2Y + 10,000) = 497,500,$$ which you can solve for$Y$and see $$Y = 162,500.$$ So, the younger son received \$162,500. We can now use either equation to find out how much the older son received. I'll use the first one to get $$162,500 + Y = 497,500$$ and so $$Y = 335,000.$$ So, the older son received \$335,000. It would not have been a problem to get a fraction/decimal for one of these values (though we didn't in this case). Banks can handle amounts of money like \$0.001, even if there isn't a coin worth so little.