Problem 367 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ 4-11(3 c-2) $$

Step-by-Step Solution

Answer

-33c + 26

Step 1 - Identify the Expression Inside the Parentheses

The given expression is: \(4 - 11(3c - 2)\). Notice the term inside the parentheses is \(3c - 2\).

Step 2 - Apply the Distributive Property

Distribute \(-11\) to both terms inside the parentheses: \[-11 \times 3c\] and \[-11 \times (-2)\]. This yields: \(-33c + 22\).

Step 3 - Simplify the Expression

Combine like terms from the distributed result with the remaining terms outside the parentheses. The simplified expression now becomes: \(4 - 33c + 22\). Combining the constants, we get: \(-33c + 26\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra means making them easier to work with. We do this by combining like terms and using properties such as the distributive property. When you simplify, the goal is to transform the expression into its simplest form without changing its value.

Let's consider the exercise given:

\[ 4 - 11(3c - 2) \]

First, we identify and simplify any expressions inside parentheses. In this example, we look at the expression inside the parentheses, which is \[ 3c - 2 \]. Once that is simplified, we can combine any like terms, which helps to make the expression more straightforward.

By breaking down these steps, you'll find it easier to handle more complex algebraic expressions!

Algebraic Distribution

The distributive property is essential in algebra. It states that multiplying a sum (or difference) by a number is the same as multiplying each addend separately and then adding the products.

Using our exercise, we need to distribute \[ -11 \] across \[ 3c - 2 \]. This gives us:
\[ -11 \times 3c = -33c \]
\[ -11 \times (-2) = 22 \]

So, after distribution, our expression becomes \[ -33c + 22 \].

Distributing is crucial because it helps to remove parentheses, simplifying the expression and making it easier to combine like terms later.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. This process makes the expression more manageable.

In our exercise, after distributing, the expression is \[ 4 - 33c + 22 \]. To simplify further, combine the constant terms: \[4 + 22 = 26 \].

Thus, the simplified expression is:
\[ -33c + 26 \]

By combining like terms, we reduce the expression to its most compact form, making solving equations and understanding the relationships between variables much easier.