Problem 70 - Algebra A Combined Function

Question

Simplify each expression. See Section 1.8. $$ 9-6(5 x+1) $$

Step-by-Step Solution

Answer

The expression simplifies to \(3 - 30x\).

Step 1: Distribute the Negative Sign

First, distribute the \(-6\) to both terms within the parentheses, which are \(5x\) and \(+1\).distribution gives us \(-6 \times 5x = -30x\) and \(-6 \times 1 = -6\).

Step 2: Rewrite the Expression

Replace the expression by substituting the distributed values back into the expression. The expression becomes: \(9 - 30x - 6\).

Step 3: Combine Like Terms

Simplify the expression by combining the constants \(9\) and \(-6\). This results in: \(3 - 30x\). This is the simplified form of the expression.

Key Concepts

Distributive Property

In algebra, the distributive property is a fundamental tool that helps in simplifying expressions. It allows you to multiply a term outside a parenthesis by each term inside the parenthesis. This is what we call distributing the term. It looks something like this: for any numbers or expressions, if you have \( a(b+c) \), you distribute \( a \) to both \( b \) and \( c \), resulting in \( ab + ac \).

For example, in the expression \( 9 - 6(5x + 1) \), you apply the distributive property by distributing the \(-6\) across the terms inside the parenthesis—that is, to \( 5x \) and \(+1\). Therefore:

• \(-6 \times 5x = -30x\)
• \(-6 \times 1 = -6\)

So, using the distributive property helps to rewrite the expression as \( 9 - 30x - 6 \). Understanding this property is essential as it allows you to break down and tackle more complex expressions with ease.

Combining Like Terms

Once you've distributed terms where needed, the next step in simplifying algebraic expressions is often combining like terms. Like terms are terms that have the same variable raised to the same power. Only these terms can be combined through addition or subtraction.

In the expression transformed by the distributive property, \( 9 - 30x - 6 \), we identify like terms. Here, the numbers \( 9 \) and \(-6\) are like terms because they are both constants without variables. Combining these numbers gives \( 9 - 6 = 3 \).

Thus, the simplified expression becomes \( 3 - 30x \). This process is crucial in algebra because it reduces expressions to their simplest form, making it easier to work with or solve them. Always look for opportunities to combine like terms to keep your expressions clear and concise.

Expression Simplification

Expression simplification is the goal of applying algebraic methods, such as the distributive property and combining like terms, to reduce expressions to their simplest form. Simplifying means making an expression as clean as possible without changing its value.

Let's take our derived expression \( 9 - 6(5x + 1) \) as an example. By using the distributive property and combining like terms, we arrived at \( 3 - 30x \). This is considered simplified because:

• All similar terms are combined.
• It’s concise and free of unnecessary complexity.
• The operations performed preserve the original expression's value.

Simplified expressions are easier to interpret and can be used to solve equations more efficiently. It's like cleaning up a sentence to make sure each word is essential. Mastering expression simplification in algebra will save time and reduce mistakes in more advanced mathematics.