91Ó°ÊÓ

When working with algebraic expressions, combining like terms is an essential step. Like terms are terms within an expression that have the same variables raised to the same powers. For example, in the expression \(2x + x\), both terms contain the variable \(x\) and can, therefore, be combined.

To combine these terms, add their coefficients (the numerical part of the term). Here, the coefficients are 2 and 1. Adding these gives us \(3x\).

Combining like terms helps in simplifying expressions and solving equations. It's critical to only merge terms that are truly 'like' (same variables and powers) to get correct results.

• Identify like terms: Same variable part
• Sum up their coefficients: Just work with the numbers

Remember, if terms are not alike, they must stay separate in the expression.

The distributive property is a key algebraic rule used to simplify expressions. This property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). In other words, you multiply the term outside the parentheses with each term inside the parentheses.

In our exercise, though it may not be obvious, the addition 'distributes' over the terms within parentheses:

• \((2x + 3) + (x - 5) = 2x + 3 + x - 5\)

Here, each term outside the parentheses directly adds to the terms inside them without changing their values, since it's positive addition.

Using the distributive property correctly makes it easier to combine and simplify expressions, and is fundamental in solving both simple and complex equations.

Simplifying algebraic expressions means reducing the expression to its simplest form, making it easier to work with. This process often involves removing parentheses through distribution, combining like terms, and ensuring no further simplification can be done.

For example, in the exercise given:

• Start with expression: \((2x+3) + (x-5)\)
• Use distributive property to remove parentheses: \(2x + 3 + x - 5\)
• Combine like terms: \(2x + x = 3x\), and \(3 - 5 = -2\)

The simplified expression is \(3x - 2\).

Simplification is crucial because it makes expressions more manageable and solutions clearer. Always check your work to ensure no further combines or reductions can be done, and ensure the simplest expression fully represents the original's operations.