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 planes. Let the speed of the San Francisco plane be \(x\) mph. Since the San Diego plane flies 50 mph slower, its speed is \(x - 50\) mph.
2. Calculate the distance each plane travels in 1/2 hour. The San Francisco plane travels \(\frac{1}{2} \times x = \frac{x}{2}\) miles, and the San Diego plane travels \(\frac{1}{2} \times (x - 50) = \frac{x - 50}{2}\) miles.
3. Since the planes are flying in opposite directions (one north, one south), the total distance between them after 1/2 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: \(\frac{x + x - 50}{2} = 275\), which simplifies to \(\frac{2x - 50}{2} = 275\).
5. Multiply both sides of the equation by 2 to eliminate the denominator, then solve the resulting linear equation for \(x\). Once \(x\) is found, calculate \(x - 50\) to find the speed of the San Diego plane.

Key Concepts

Relative Distance and Speed

When two objects move in opposite directions, the distance between them increases at the sum of their speeds. Understanding how to combine speeds to find relative distance is essential for setting up equations involving moving objects.

Formulating Algebraic Equations from Word Problems

Translating the problem's conditions into algebraic expressions is crucial. Assign variables to unknown quantities, express relationships (like one speed being 50 mph slower), and use given data to form equations that can be solved systematically.

Solving Systems of Equations

Problems with multiple unknowns often require solving systems of equations. Techniques such as substitution or elimination help find the values of variables that satisfy all given conditions simultaneously.