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The distributive property is a key concept that helps in simplifying algebraic expressions, especially when dealing with parentheses. It states that for any numbers or variables, distributing a multiplier across terms inside a parenthesis is equivalent to multiplying each term separately and then adding the results. For example, to simplify the expression \(7(3y - 5)\), you apply the distributive property by multiplying 7 with each term inside the parenthesis:

• Multiply \(7\times 3y\) to get \(21y\).
• Multiply \(7\times -5\) to get \(-35\).

This distributes the 7 across each internal term. Similarly, for the expression \(2(4y + 3)\), distribute 2 to get:

• Multiply \(2\times 4y\) to get \(8y\).
• Multiply \(2\times 3\) to get 6.

The result of applying the property in both cases yields a simpler expression, which can be further simplified by combining like terms. This technique is a basic yet powerful tool in algebra for managing more complex expressions.

After using the distributive property, the next step in simplifying algebraic expressions is to combine like terms. Like terms are terms that have the same variable raised to the same power. In the expression \(21y - 35 + 8y + 6\), the like terms can be identified and combined:

• Combine the 'y' terms: \(21y + 8y\). Adding these together gives \(29y\) because they both contain the same variable 'y'.
• Combine the constant terms: \(-35 + 6\). When added together, this equals \(-29\).

Thus, the expression simplifies to \(29y - 29\). Combining like terms is vital for keeping expressions manageable and for finding the simplest form of an expression. It involves careful attention to ensure only like terms are combined, and it is an essential skill when working with algebraic equations.

An algebraic expression is a mathematical phrase that can contain numbers, variables, and arithmetic operations, such as addition, subtraction, multiplication, and division. They do not have an equality sign like algebraic equations do. The expression from our example, \(7(3y - 5) + 2(4y + 3)\), is an algebraic expression.These expressions are used in various branches of mathematics to represent problems or to model real-world situations. Understanding how to work with them, such as by applying the distributive property or combining like terms, is crucial. It allows for simplification and manipulation, facilitating problem-solving and ensuring expressions are presented in their simplest form.The key to mastering algebraic expressions is practicing these techniques across different scenarios and understanding how each term contributes to the whole. By doing so, a student can more readily manage complex expressions and equations in both academic and practical applications.