Solve each problem. See Example 2. Two planes leave Los Angeles at the same time. One heads south to San Diego, while the other heads north to San Francisco. The San Diego plane flies 50 mph slower than the San Francisco plane. In 1/2 hr, the planes are 275 mi apart. What are their speeds?

Verified step by step guidance

1. Define variables for the speeds of the two planes. Let the speed of the San Francisco plane be \(x\) mph. Since the San Diego plane flies 50 mph slower, its speed will be \(x - 50\) mph.
2. Express the distances each plane travels in terms of their speeds and the time traveled. Both planes travel for \(\frac{1}{2}\) hour, so the distance traveled by the San Francisco plane is \(x \times \frac{1}{2} = \frac{x}{2}\) miles, and the distance traveled by the San Diego plane is \((x - 50) \times \frac{1}{2} = \frac{x - 50}{2}\) miles.
3. Since the planes are flying in opposite directions, the total distance between them after half an hour is the sum of the distances each has traveled. Set up the equation: \(\frac{x}{2} + \frac{x - 50}{2} = 275\).
4. Combine like terms on the left side of the equation to simplify it. This will give you an equation in terms of \(x\) that you can solve.
5. Solve the resulting linear equation for \(x\) to find the speed of the San Francisco plane. Then subtract 50 from \(x\) to find the speed of the San Diego plane.

Key Concepts

Relative Distance and Speed

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. The total distance between them after a certain time is the product of this relative speed and the time elapsed.

Setting Up Algebraic Equations

Translate the word problem into algebraic expressions by defining variables for unknown quantities. Use given relationships, such as one speed being slower by a certain amount, to form equations that can be solved systematically.

Solving Systems of Equations

Use substitution or elimination methods to solve the system of equations derived from the problem. This process finds the values of unknown variables, such as the speeds of the planes, that satisfy all given conditions.