91Ó°ÊÓ

Algebraic expressions are mathematical phrases that involve variables, numbers, and operations. In our original exercise, the expression to simplify is:

• 9(z + 2) + 5(3 - z)

This expression contains variables ('z'), numbers, and mathematical operations (addition and multiplication).
Algebraic expressions can take many forms, like sums, differences, and products. They do not include equal signs or inequality symbols, which differentiate them from equations or inequalities.
Understanding the structure of these expressions is crucial for simplification. By identifying the components, we can apply mathematical rules to manipulate them. This manipulation clarifies the expression into its simplest form. This process often involves using other concepts like the distributive property and combining like terms.

The distributive property is a key algebraic rule used to simplify expressions. It states that, for any numbers or variables, the expression \(a(b + c)\) is equal to\(ab + ac\).
In our example, the distributive property is applied twice:

• First, with \(9(z + 2)\) giving \(9z + 18\).
• Then, with \(5(3 - z)\) resulting in \(15 - 5z\).

The power of the distributive property lies in simplifying expressions containing parentheses. It helps expand the expression to show each term separately.[
By understanding and using this property, you'll simplify each part efficiently before combing all into a single expression. It breaks down complex problems into easier steps, making algebra less intimidating.

Combining like terms is the final step for simplifying algebraic expressions efficiently.
What are like terms? They contain the same variable or no variable at all, and are raised to the same power. The only difference lies in their coefficients.
In the exercise provided:

• Like terms include the \(z\) terms in \(9z - 5z\),
• as well as constant numbers together: \(18 + 15\).

Combining these terms results in: \(4z + 33\).This means adding their coefficients. For the \(z\) terms, subtracting: \(9 - 5 = 4\).For the constants, add: \(18 + 15 = 33\).
Combining like terms reduces expressions into simpler, more manageable forms. Identifying and grouping similar entities streamlines the expression into a clearer solution, making it easier to understand and use in further calculations.