The price of a stereo at Mary's is $220 less than twice the price at Cal's. The difference between the stores is $175. How much does the system cost at each store? Can you show how to write the equation and solve it?

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Essentially, there are two unknowns: for the price at Marys, denote it as `x` (dollars), and the price at Cal's as `y` (dollars).

We are given two statements about these prices. The first statement, "the price of a stereo at Mary's is $220 less than twice the price at Cal's", gives us the first equation, `x = 2 * y - 220.` The second one, "the difference between the stores is $175", gives us the second equation, `| x - y | = 175.` I used absolute value because it is not clear which price is subtract from which. Suppose that we subtract the smaller price from the greater.

There is a system of two equations with two unknowns, `x = 2y - 220` and `| x - y | = 175.` To solve it, substitute `x` from the first equation to the second and obtain `|2y - 220 - y| = 175,` `|y - 220| = 175.`

This equation has two solutions. For `y gt= 220` we get

`|y - 220| = y - 220 = 175, y = 175 + 220 = 395 ( $ ) .`

For `y lt 220` we get

`|y - 220| = 220 - y = 175, y = 220 - 175 = 45 ( $ ) .`

Now find the corresponding x's: for the first case, `x = 2 y - 220 = 790 - 220 = 570 ( $ ) .` For the second case, `x = 2 y - 220 = 90 - 200 = - 110 ( $ ).` A negative price has no sense for our problem, so we discard this solution.

The answer is: the price at Marys is 570$, the price at Cal's is 395$.

If you definitely want to write exactly one equation, we can pretend there is only one unknown, the price at Cal's, which is y . Then the price at Mary's is 2 y - 220 and we obtain "one" equation |y - 220| = 175.

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