A farmer has 150 feet of fence available to enclose a 1125 square foot region in the shape of adjoining squares with sides of length X&Y. The big square has side of length X and the small square has side of length Y. Find X&Y. Well, if we start at a corner and go around this figure to get our perimeter, let's say we start here. We know that this length here is X and this length here is Y. Here is another Y coming down is AY. Now this piece here is actually the whole X length minus what we had already used of the Y. This piece here is an X and this piece here is an X. We thought about this whole length here being X and then what we've already used at the top was AY. So if we Add all those together we get 4X plus 2Y equal 150 and the sum of the areas. So X ^2 + y ^2 has to equal a total of 1125. If we take that linear equation and solve for Y, we get Y equal negative two X + 75. If we then substitute that into the equation for the circle or the area formula X ^2 plus the quantity negative two X + 75 ^2 equaling 1125. If we distribute it out and take everything to one side, so we have zero on one side, and then I'm going to simplify dividing by 5. So I get X ^2 - 60, X plus 900 = 0. Then we're going to factor that X -, 30 X -30 so X equal 30. And if X is 30, we're going to substitute that into the Y equation. So Y equal -2 * 30 + 75 or Y is 75 S 30 feet for the one square and 70 and 15 feet for the 2nd.