### Instruction: If $725x + 727y = 1500$ and $729x+ 731y = 1508$, what is the value of $x - y$ ?### Response: I notice that the coefficients of $x$ and $y$ are very close to each other in both equations, so I suspect that eliminating one of the variables by subtraction might be a good idea.
If I subtract the first equation from the second, I get $(729x + 731y) - (725x + 727y) = 1508 - 1500$, which simplifies to $4x + 4y = 8$.
Dividing both sides by 4, I get $x + y = 2$.
Now I want to find another equation that relates $x$ and $y$, but in a way that allows me to find $x - y$.
I notice that if I add the first equation to the second, I get $(725x + 727y) + (729x + 731y) = 1500 + 1508$, which simplifies to $1454x + 1458y = 3008$.
Dividing both sides by 2, I get $727x + 729y = 1504$.
Now I have two equations: $x + y = 2$ and $727x + 729y = 1504$.
If I subtract the first equation from the second, I get $(727x + 729y) - (x + y) = 1504 - 2$, which simplifies to $726x + 728y = 1502$.
Dividing both sides by 2, I get $363x + 364y = 751$.
Now I have two equations: $x + y = 2$ and $363x + 364y = 751$.
If I multiply the first equation by 363, I get $363x + 363y = 726$.
If I subtract this equation from the second, I get $(363x + 364y) - (363x + 363y) = 751 - 726$, which simplifies to $y = 25$.
Now that I have $y$, I can plug it into the first equation to get $x + 25 = 2$, which implies that $x = -23$.
Therefore, $x - y = -23 - 25 = -48$.