91Ó°ÊÓ

Speed and distance problems are a common type of question in algebra that help us understand how objects move in different conditions. Here, we are dealing with a light aircraft flying with and against the wind. These types of problems connect speed, distance, and time, using the basic formula:

• Distance = Speed × Time

In this example, the aircraft covers a distance of 330 miles. When flying with the wind, the wind's speed adds to the aircraft’s speed, allowing it to travel faster. Conversely, when flying against the wind, the wind’s speed subtracts from the aircraft’s speed, slowing it down. By analyzing how the aircraft performs in each scenario, we can solve for unknowns such as the wind speed, as we do in this exercise.

A system of equations is a set of two or more equations with the same variables. Solving them helps us to find values for these variables simultaneously. In our problem:

• We define two variables: x (speed of the aircraft in still air) and y (speed of the wind).
• We create two equations based on the conditions given in the problem.

The equations are:\[ 2(x + y) = 330 \]\[ 3(x - y) = 330 \]These equations arise from how the aircraft behaves when flying with and against the wind. By using algebraic techniques like substitution or elimination, we can solve these equations to find the values of x and y. This process reflects the essence of working with systems of equations, enabling us to handle complex problems with multiple variables.

Simplifying equations is a fundamental step in solving systems of equations. It involves reducing equations to their simplest form so they are easier to work with. In our example:

• We start with the equations derived from the problem.
• By dividing both sides of each equation, we simplify them.

The simplifications are:\[ x + y = 165 \]\[ x - y = 110 \]By simplifying, we transform complex, detailed equations into straightforward ones that are easier to solve. Simplification helps us isolate variables, apply methods such as addition or substitution, and ultimately find the solution efficiently. This process underscores the elegance and power of algebra in tackling real-world problems.