Problem 59 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ -9(4 x+8)+2 $$

Step-by-Step Solution

Answer

The expression simplifies to \(-36x - 70\).

Step 1: Apply the Distributive Property

To remove the parentheses in the expression \[-9(4x + 8) + 2\], begin by distributing the \(-9\) across each term inside the parentheses: \[-9 \cdot 4x + (-9) \cdot 8\].This yields:\[-36x - 72\].

Step 2: Rewrite the Expression

Substitute the distributed expression back into the original expression:\[-36x - 72 + 2\].

Step 3: Simplify the Expression

Combine like terms, which are the constant terms:\[-72 + 2\] simplifies to \(-70\).Thus, the expression simplifies to:\[-36x - 70\].

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra. It involves transforming complicated expressions into their simpler, more manageable forms. This process helps in obtaining the solution more efficiently. To simplify an expression, you may need to apply different algebraic rules such as the distributive property, combining like terms, and rearranging terms.

In our given example, we needed to simplify the expression \(-9(4x + 8) + 2\). Simplification began with the use of the distributive property to remove parentheses, multiplying each term inside the parentheses by \(-9\). This gave us \(-36x - 72\).

The next step was to manage the entire expression by writing it as \(-36x - 72 + 2\). Then, we simplified it further by combining the constant terms: \(-72 + 2\), resulting in \(-70\). The final simplified expression is \(-36x - 70\).

Simplifying can make evaluation and solving equations much easier.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions. It involves merging terms that have the same variable component—or no variable at all—by adding or subtracting their coefficients.

In the expression \(-36x - 72 + 2\), we identified like terms: the constants \(-72\) and \(2\). We then combined these constants by performing the arithmetic \(-72 + 2\), equaling \(-70\). This resulted in the simplified expression: \(-36x - 70\).

To find like terms, look for variables that are raised to the same power and combine them with standard arithmetic operations. This step is essential to ensuring your algebraic expressions are as compact as possible for easier interpretation or further mathematical manipulation.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Expressions do not contain equality signs, unlike equations, and thus do not show a complete relationship.

In the context of our exercise, \(-9(4x + 8) + 2\) is an algebraic expression. It involves a combination of numbers, variable terms, and operations (multiplication, addition, and subtraction).

Understanding the components of algebraic expressions is essential when applying algebraic rules such as the distributive property to simplify them. Simplifying algebraic expressions helps in revealing underlying patterns and relationships, facilitating easier handling of mathematical challenges.