Problem 123 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -(c+7)+2(c-3) $$

Step-by-Step Solution

Answer

The simplified expression is \(c - 13\).

Step 1: Distribute the Negative Sign

Start by distributing the negative sign over the parentheses in the expression \(-(c+7)\). This will change \(c+7\) into \(-c-7\). The expression becomes: \[-c - 7 + 2(c-3)\]

Step 2: Distribute the 2

Distribute the 2 through the expression inside the parentheses, \(2(c-3)\). This gives \(2c - 6\). Replace it in the expression: \[-c - 7 + 2c - 6\]

Step 3: Combine Like Terms

Now, combine the like terms in the expression. Combine \(-c\) and \(2c\) to get \(c\), and combine \(-7\) and \(-6\) to get \(-13\). This simplifies the expression to: \[c - 13\]

Step 4: Conclusion

The simplified form of the given expression \(-(c+7) + 2(c-3)\) is \(c - 13\).

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra used to simplify expressions. When you see an expression with a term outside the parentheses, like \(a(b + c)\), the distributive property allows you to multiply that term by each term inside the parentheses.
Let's break it down further:

• Imagine you have \(a imes (b + c)\).
• Using the distributive property, this becomes \(a imes b + a imes c\).

This step is crucial when simplifying expressions, as it helps eliminate the parentheses, making it easier to work with the expression.
In this exercise, we distributed the -1 (negative sign) over the expression inside the parentheses, transforming \(-(c+7)\) into \(-c-7\). This operation is known as distributing a coefficient over the terms inside the parentheses.

Combining Like Terms

Combining like terms is an important technique in algebra that helps to simplify expressions by merging terms that have the same variables raised to the same power.
To combine like terms, follow these guidelines:

• Identify terms that have the same variable and exponent.
• Add or subtract the coefficients of these terms.

For example, consider the expression \(-c - 7 + 2c - 6\).
Here, \(-c\) and \(+2c\) are like terms because both terms involve the variable \(c\) raised to the first power.
Subtracting and adding their coefficients, we combine them to get \(c\). On the other hand, \(-7\) and \(-6\) are also combined because they are constant terms without variables, and when added, they give \(-13\).
This results in a simpler expression: \(c - 13\).

Negative Sign Distribution

Negative sign distribution can seem tricky initially, but it is a straightforward application of the distributive property. In algebra, when you have a negative sign in front of a set of parentheses, it means you need to distribute \(-1\) to each term inside the parentheses.
Here's how you handle it:

• If you have \(-(a + b)\), it becomes \(-a - b\).
• This happens because \(-1 imes a = -a\) and \(-1 imes b = -b\).

In the original problem, we had \(-(c + 7)\). Distributing the negative sign gave us \(-c - 7\).
This step is essential for simplifying expressions, as it allows you to dismantle complex groupings and reorganize them into more manageable terms.