Problem 33 - 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. $$ 9(x-6) $$

Step-by-Step Solution

Answer

9x - 54

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). Here we have \(9(x - 6)\). We will multiply \(9\) by each term inside the parentheses. This gives us: \(9 \times x - 9 \times 6\).

Step 2: Perform the Multiplication

Now perform the multiplications: \(9 \times x = 9x\) and \(9 \times 6 = 54\). Substitute these values back into the equation to get \(9x - 54\).

Step 3: Simplify the Expression (If Needed)

The expression \(9x - 54\) is already simplified, as there are no like terms that can be combined. Therefore, this is the simplest form of the expression.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. Unlike a simple number, they can represent a range of values based on the variables they contain. In the expression \(9(x-6)\), "\(x\)" is a variable that can take on different values. An expression can consist of:

• Constants, which are fixed numbers like 9 and 6 in our example.
• Variables, which are symbols that represent unknown values; here it's \(x\).
• Operators, such as addition and subtraction, which show relationships between constants and variables.

The goal when working with algebraic expressions is often to simplify them by combining like terms or applying properties such as the distributive property.

Simplifying Expressions

Simplifying an expression involves reducing it to its most compact, easily understandable form. This process might involve combining like terms, removing parentheses, or using mathematical properties to make the expression shorter and more straightforward.
In the expression \(9x - 54\), after applying the distributive property, there are no like terms to simplify further. Like terms are terms whose variables (and their exponents) are the same, which is why \(9x\) and \(-54\) cannot be combined.
Sometimes simplification can also involve factoring out common factors or rearranging terms to highlight certain parts of an expression. However, in this case, the expression is already in its simplest form.

Multiplication in Algebra

Multiplication in algebra often involves working with variables and constants together. Understanding how to perform multiplication correctly is a fundamental part of simplifying complex algebraic expressions.
When you multiply a constant by a variable as in \(9 \times x\), you get a term like \(9x\), which is shorthand for "9 times the value of \(x\).”
Here are the key steps in our multiplication process:

• Apply the Distributive Property: Distribute the 9 across each term inside the parentheses: \(9(x - 6)\).
• Multiply Each Term: Compute \(9 \times x = 9x\) and \(9 \times 6 = 54\).

This ensures that every part of the expression inside the parentheses is multiplied correctly by the outside term. It's crucial in keeping expressions accurate and balanced, which is especially important in solving equations.