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An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols like addition, subtraction, multiplication, and division. It's like a recipe that represents a mixture of different ingredients, where variables are unknown quantities and coefficients are the numbers that multiply these variables. For example, in the expression \( -33.9 xy + 58.2 xy \) from the exercise, \(xy\) is the variable portion and \( -33.9 \) and \( 58.2 \) are the coefficients.

Understanding algebraic expressions is crucial because they form the foundation of algebra and are essential tools for modeling real-life situations. They allow us to describe relationships and changes that occur in different scenarios, like calculating the speed of a car or predicting the growth of a population.

The process of simplifying expressions involves reducing an algebraic expression to its most basic form without changing its value. This is done to make the expression easier to understand and work with. There are several methods to simplify expressions, and one such method is combining like terms.

For instance, in the solution to our exercise, we combined \( -33.9 xy \) and \( 58.2 xy \) into \( 24.3 xy \) to simplify the original expression. Simplifying can involve other operations as well, such as distributing, factoring, or canceling terms, but the goal is always to reduce complexity while maintaining the expression's true nature.

When we talk about like terms, we're referring to terms within an algebraic expression that have the same variables raised to the same power. These can be combined using addition or subtraction. In our exercise, \( -33.9 xy \) and \( 58.2 xy \) are like terms because they both contain the product of \(x\) and \(y\). We combined these by adding their coefficients to get \( 24.3 xy \).

It's important to remember that only like terms can be combined. If the variables or their powers don't match, such as \(x\) and \(x^2\), or \(xy\) and \(xz\), then they cannot be combined through addition or subtraction. Recognizing like terms is a vital skill because it helps in simplifying expressions and solving equations effectively.