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In algebra, 'like terms' are terms that contain the same variables, with each variable raised to the same power. For example, in the expression \(-5x - 10y - 3x + 13y\), the like terms are \(-5x\) and \(-3x\), as well as \(-10y\) and \(13y\). Each pair of terms includes the same variable, either \(x\) or \(y\), making them eligible for combination through addition or subtraction.

Understanding and identifying like terms is crucial as it sets the stage for simplifying complex algebraic expressions. Every term in an expression is either a constant or a coefficient multiplied by a variable(s), and only those with the same variable parts can be combined. This simplifies problems, reducing potential errors and making calculations quicker. Whenever you're working with algebraic expressions, begin by analyzing which terms are alike.

Simplification is the process of reducing an algebraic expression to its simplest form by performing all possible operations. This is usually done by combining like terms and eliminating any redundancies. In our example, starting with \(-5x - 10y - 3x + 13y\), the simplification occurs by recombining the variables accordingly.

After identifying like terms, the next step involves performing operations to combine them. Simplification aims to make expressions more manageable and easier to understand, which is helpful for solving equations or functions. Consider simplification as a kind of decluttering—getting rid of unnecessary complexity to focus on what matters the most. It not only decreases the risk of errors but also enhances the communication of mathematical ideas.

Combining like terms is a fundamental step in the simplification process that involves adding or subtracting coefficients of terms with the same variable part. For instance, in the expression \(-5x - 3x\), both terms are like terms, since they involve the \(x\) variable. Combining them, \(-5x + (-3x)\) results in \(-8x\). Similarly, combining \(-10y + 13y\) gives \(3y\).

When you combine like terms, it is purely an arithmetic operation on coefficients. The variable and its power do not change. This step simplifies expressions and is key to solving equations or further algebraic manipulation. A pro tip is to keep an eye on the signs (positive or negative) of each term so as to ensure accurate combination. Reducing expressions by properly combining like terms can make solving equations a smoother and quicker process.