Basic Math Word Problems

From (x - y = 3), we can express (x) as (x = y + 3). Substituting this into (x^2 + y^2 = 25), we get ((y + 3)^2 + y^2 = 25). Expanding and simplifying, we get (2y^2 + 6y + 9 = 25). Subtracting 9 from both sides, we get (2y^2 + 6y - 16 = 0). Factoring, we get ((2y - 4)(y + 4) = 0). Therefore, (y = 2) or (y = -4). If (y = 2), then (x = y + 3 = 2 + 3 = 5). Therefore, (x + y = 5 + 2 = 7). If (y = -4), then (x = y + 3 = -4 + 3 = -1). Therefore, (x + y = -1 + (-4) = -5). Since (x + y) cannot be negative, the correct answer is (7).

The farmer plants corn on (120 \times \frac{2}{5} = 48) acres of land. He plants soybeans on (120 \times \frac{1}{3} = 40) acres of land. Therefore, he plants wheat on (120 - 48 - 40 = 32) acres of land.

The first train travels for 1 hour before the second train leaves. In that time, it travels (60 \times 1 = 60) miles. Therefore, the distance between the two trains at 11:00 AM is (900 - 60 = 840) miles. The two trains are traveling towards each other at a combined speed of (60 + 70 = 130) miles per hour. Therefore, they will meet in (840 \div 130 = 6.5) hours. Since the first train leaves at 10:00 AM, the two trains will meet at 10:00 AM + 6.5 hours = 4:30 PM.

There are (100 \times 0.60 = 60) women and (100 \times 0.40 = 40) men in the company. There are (60 \times 0.20 = 12) women managers and (40 \times 0.30 = 12) men managers. Therefore, the company has a total of (12 + 12 = 24) managers.

The area of a rectangle is given by (A = l \times w). Therefore, (100 = x \times y). The perimeter of a rectangle is given by (P = 2l + 2w). Substituting (x = \frac{100}{y}) into the formula for the perimeter, we get (P = 2 \left(\frac{100}{y}\right) + 2y = \frac{200}{y} + 2y = \frac{200 + 2y^2}{y}). Therefore, the perimeter of the garden is (\frac{200 + 2y^2}{y}).