Problem 63 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ -12 x-5(11-x) $$

Step-by-Step Solution

Answer

The simplified expression is -7x - 55

Step 1: Apply the Distribution Law

Apply the distributive law to the expression -5(11 - x). -5 multiplies both 11 and -x resulting in -55 + 5x.

Step 2: Combine Like Terms

Combine the similar terms from the expression resulted in Step 1 (-55 + 5x) with -12x. Hence, -12x - 55 + 5x.

Step 3: Simplify the Expression

Now, combine the like terms -12x and 5x. This results in -7x -55.

Key Concepts

Simplification

Simplification of algebraic expressions involves making the expression as concise as possible while keeping the same value. Let’s break it down!
The goal is to convert complex expressions into simpler forms, which makes them easier to work with or evaluate. Here's the process:

• Simplification may involve: applying arithmetic operations, eliminating parentheses by distributing when necessary, and combining like terms.
• This process reduces chances for calculation mistakes and makes further algebraic manipulations manageable.
• For instance, consider the expression from our example: \(-12x - 5(11-x)\). We start by distributing and then combine like terms.

Simplifying expressions is particularly useful as it lays the groundwork for solving equations and inequalities.

Distributive Property

The distributive property is a crucial concept in algebra that allows us to multiply a single term with terms inside a parenthesis. Let’s explore this concept in the context of the expression: \(-5(11 - x)\).

• The distributive property states that \( a(b + c) = ab + ac \). This means you multiply the term outside the parenthesis (here, -5) by each term inside the parenthesis. So, \(-5 \times 11 = -55\) and \(-5 \times -x = 5x\).


• Keep careful track of signs: Multiplying a negative number by another negative (\(-5 \, \text{and} \, -x\)) results in a positive number \(5x\).

By distributing, we transform our expression into an equivalent one that’s easier to handle (-55 + 5x). This step is vital before we combine like terms.

Combining Like Terms

Combining like terms is perhaps one of the most important steps in simplifying algebraic expressions. In our task, we deal with terms like \(-12x\) and \(+5x\).

• "Like terms" are terms that contain the same variable raised to the same power. You can add or subtract them directly, which is what happens with these x-terms.For our expression (-12x + 5x), simply combine them: \(-12x + 5x = -7x\).


• This operation reduces the number of terms and simplifies the expression, making it easier to use in further calculations.Remember, you can only combine the coefficients of like terms, leaving the variable part unchanged.

After combining these terms, the final simplified expression becomes \(-7x - 55\), succinct and ready for further algebraic processes or evaluations.