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The distributive property is a fundamental concept in algebra that allows us to simplify expressions by distributing multiplication over addition or subtraction. Imagine you have an expression like the one in our exercise: \(3(2x - 5)\). The distributive property tells us to multiply every term inside the parentheses by 3. So, \(3 \times 2x - 3 \times 5\) becomes \(6x - 15\).

Similarly, when you have a negative or minus sign before the parentheses, such as with \(-5(x - 4)\), you must distribute the -5 across both terms: \(-5 \times x + 5 \times 4\), resulting in \(-5x + 20\). By consistently applying the distributive property, you break down expressions into simpler components, making them easier to work with and solve.

Use this property to handle complex expressions in manageable pieces, ensuring each term is multiplied as necessary.

Combining like terms is an important step in simplifying expressions. Terms in an algebraic expression are "like" terms if they have the same variable raised to the same power. For instance, \(6x\) and \(-5x\) are like terms because they both contain the variable \(x\) raised to the power of 1.

In our exercise, after distributing, we have \(6x - 5x\). To combine these like terms, you sum or subtract their coefficients: \(6 - 5 = 1\). Therefore, \(6x - 5x\) simplifies to \(1x\), which we usually write simply as \(x\).

Similarly, constants are also considered like terms. Take \(-15\) and \(20\): you combine them by performing the arithmetic operation \(-15 + 20 = 5\).

By combining like terms, you reduce expressions to their simplest form, helping you solve equations more efficiently.

Simplifying expressions is the final goal in our problem-solving process. Once you've distributed multiplication across addition or subtraction and combined like terms, you should end up with the simplest form of the expression.

In our example, after completing the distribution, our expression \(6x - 15 - 5x + 20\) becomes \(x + 5\) after combining the like terms. This is the simplest form of the expression.

When simplifying an expression, always ensure you:

• Use the distributive property to eliminate parentheses.
• Identify and combine like terms to streamline the expression.
• Perform any additional arithmetic like adding or subtracting constant values.

This process helps not only in solving algebraic problems effectively but also in ensuring clarity and error reduction in mathematical operations.