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The distributive property is a fundamental concept in algebra. It is used to simplify expressions and is especially valuable when you're dealing with parentheses. The property states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. For example, if you have an expression like \(2(a + 2b)\), you apply the distributive property by multiplying 2 with each term inside the parenthesis:

• Multiply 2 by \(a\), giving you \(2a\).
• Multiply 2 by \(2b\), resulting in \(4b\).

By doing this, you've expanded the expression into \(2a + 4b\). This is the same principle we applied to simplify the first part of our original expression. Using the distributive property helps break down more complicated expressions into simpler, individual components that are easier to manage.

Once you have expanded your expression using the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients. For instance, in the expression \(2a + 4b - a + 3b\), you can find terms with the same variables:

• The terms \(2a\) and \(-a\) both contain the variable \(a\).
• The terms \(4b\) and \(3b\) both contain the variable \(b\).

To combine them:

• Add \(2a\) and \(-a\) to get \(a\).
• Add \(4b\) and \(3b\) to get \(7b\).

The process of combining like terms simplifies the expression further by reducing the number of terms, making it easier to interpret and utilize in solving or further simplifications.

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They serve as the foundation of algebra and appear in countless algebraic operations and equations. In its essence, an algebraic expression like \(a + 7b\) represents different quantities or values. The terms consist of a coefficient (a constant) multiplied by a variable (like \(a\) or \(b\)).Understanding algebraic expressions is crucial:

• They allow you to describe real-world situations in mathematical terms.
• Serve as a basis for constructing algebraic equations by equating them to other expressions or constants.
• Facilitate the solving of problems by allowing manipulation via algebraic rules and properties.

The simplified expression \(a + 7b\) is cleaner and more manageable to work with for further operations or to substitute with known values to solve real-world problems. Mastery of expressing and simplifying algebraic expressions opens the doors to deeper mathematical understanding and more complex problem-solving.