Problem 41 - Prealgebra

Question

Simplify expression. \(c+2(d-5 c)\)

Step-by-Step Solution

Answer

The simplified expression is \(-9c + 2d\).

Step 1: Distribute 2 into the Parentheses

The expression is given as \( c + 2(d - 5c) \). Begin by distributing the 2 across the terms inside the parentheses. This means we multiply 2 by each term inside the parentheses: \( 2 \times d \) and \( 2 \times (-5c) \). The expression becomes \( c + 2d - 10c \).

Step 2: Combine Like Terms

After distributing, the expression is \( c + 2d - 10c \). Now, combine the like terms. Like terms are terms that have the same variable raised to the same power. Here, the like terms are \( c \) and \(-10c \). Combine them as follows: \( c - 10c = -9c \).

Step 3: Write the Simplified Expression

After combining the like terms, the expression simplifies to \( -9c + 2d \). This is the simplest form of the given expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions involving parentheses. It allows us to "distribute" a factor across terms inside a parenthesis. This means you will multiply the outer term by each term within the parenthesis. For example, in the expression \( 2(d - 5c) \), the 2 is distributed to both \( d \) and \( -5c \).

Thus, you perform the multiplication:

• \( 2 \times d \) gives you \( 2d \).
• \( 2 \times (-5c) \) results in \( -10c \).

By applying the distributive property, the expression transforms into \( c + 2d - 10c \). This process sets the stage for the next step in simplifying the expression. Understanding how to appropriately use the distributive property is a key skill for anyone working with algebraic expressions.

Combining Like Terms

Once you've completed the distribution, the next step is to simplify the expression further by combining like terms. 'Like terms' refer to terms that have identical variable parts; they have the same variables raised to the same powers. In our simplified expression \( c + 2d - 10c \), the like terms are \( c \) and \( -10c \).

These terms can be combined by simply adding or subtracting their coefficients:

• The term \( c \) is essentially \( 1c \), so when you subtract \( 10c \), you get \( 1c - 10c = -9c \).

The other term, \( 2d \), does not have any like terms to combine with, so it remains as it is.

By combining like terms, the expression shortens to \( -9c + 2d \), making it more straightforward and easier to interpret.

Algebraic Expressions

Algebraic expressions are a way of representing numbers and operations using symbols and variables. Expressions like \( c + 2(d - 5c) \) are composed of numbers, variables (like \( c \) and \( d \)), and arithmetic operations.

These expressions are at the core of algebra, providing a foundation for modeling real-world situations and solving mathematical problems. When you simplify an algebraic expression:

• You aim to make it as compact as possible.
• You ensure it is easier to understand and work with.

Simplifying often involves techniques like using the distributive property and combining like terms, as we've seen.

Mastering these concepts requires practice but pays off through better understanding and efficiency in solving algebraic problems.