Problem 93 - Beginning and Intermediate Algebra

Question

Write each expression without parentheses. $$ -(-5 c-4 d) $$

Step-by-Step Solution

Answer

5c + 4d

Step 1 - Distribute the negative sign

To remove the parentheses, distribute the negative sign through the expression \(-(-5c - 4d)\). Multiply each term inside the parentheses by -1.

Step 2 - Simplify each term

Multiply the individual terms: \(-1 \cdot (-5c) = 5c\) \(-1 \cdot (-4d) = 4d\).

Step 3 - Write the simplified expression

Combine the results from Step 2 to find the simplified expression without parentheses: \(5c + 4d\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in algebra. It makes complex problems easier to solve. The goal is to break down an expression into its simplest form.
When simplifying an expression, you want to perform operations such as distribution, combining like terms, and removing unnecessary parentheses.
This helps us better understand and solve problems. Let's look at how it works with the example \( -(-5c - 4d) \).
By distributing the negative sign (multiplying by \(-1\)), we simplify \( -(-5c - 4d) \) to \( 5c + 4d \).

Parentheses in Algebra

Parentheses are used in algebra to indicate which operations should be performed first. They group parts of an expression to show that these should be treated as one unit.
When you see parentheses, pay close attention to what is inside them. Often, you will need to simplify or distribute terms before dealing with other parts of the expression.
In our example, \(-(-5c - 4d)\), the outer negative sign must be distributed through the entire expression inside the parentheses.
This involves multiplying each term by \(-1\), leading us to remove the parentheses altogether.

Negative Multiplication

Multiplying by a negative number changes the sign of the number you are multiplying. When you multiply two negative numbers, the result is positive.
This rule is essential in algebra, especially when simplifying expressions involving parentheses. In our example, \(-(-5c - 4d)\), the multiplication of \(-1\) and \(-5c\) gives us \(5c\). The same happens for \(-1\) and \(-4d\), leading to \(4d\).
Understanding how negative multiplication works helps you correctly simplify expressions involving negative signs and parentheses.