Problem 88 - Elementary and Intermediate Algebra

Question

Simplify. $$ 12(m+11)-11+m $$

Step-by-Step Solution

Answer

The simplified expression is \(13m + 121\).

Step 1: Distribute the Constants

In the expression, distribute the constant 12 across the terms inside the parentheses. The expression becomes: \[ 12 \times m + 12 \times 11 - 11 + m \]which simplifies to: \[ 12m + 132 - 11 + m \]

Step 2: Combine Like Terms

Identify and combine like terms in the expression. Like terms are those with the same variable or constants solely. - Combine the terms with \(m\): \[12m + m = 13m\]- Combine the constant terms: \[132 - 11 = 121\]This simplifies the expression to: \[13m + 121\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps to simplify expressions by eliminating parentheses. It allows us to multiply a single term across each term inside a bracket, effectively spreading the multiplication over addition or subtraction. This is done by using the formula: \[a(b + c) = ab + ac\]In our exercise, the distributive property is applied to simplify the expression \(12(m + 11) - 11 + m\). Here, the number 12 is distributed across each term within the parentheses, leading to:- \(12 \times m = 12m\)- \(12 \times 11 = 132\)Thus, the expression becomes: \[12m + 132 - 11 + m\]Utilizing the distributive property helps to restructure and simplify the expression, making it easier to handle in subsequent steps.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms whose variables (and their respective exponents) are the same. For example, \(3x\) and \(5x\) can be combined, but \(3x\) and \(5y\) cannot since their variables are different. Additionally, constant numbers without variables are also like terms.In the expression \(12m + 132 - 11 + m\), we have two groups of like terms:

• Terms with \(m\): \(12m\) and \(m\)
• Constant terms: 132 and -11

To combine them, we simply add or subtract the numerical coefficients:- Combine \(m\)-terms: \(12m + m = 13m\)- Combine constants: \(132 - 11 = 121\)By combining these like terms, we achieve a more concise expression: \(13m + 121\). This step reduces the complexity of the expression without changing its value.

Expression Simplification

Expression simplification is an essential skill in algebra that focuses on rewriting expressions in their simplest form. Simplifying expressions involves several steps, including applying the distributive property and combining like terms, as shown in the original exercise.Here are the key steps in simplifying the expression \(12(m+11)-11+m\):1. **Distribute the constant:** Multiply each term within the parentheses by 12, resulting in \(12m + 132\).2. **Reorganize the expression:** Add the remaining parts, giving \(12m + 132 - 11 + m\).3. **Combine like terms:** Group and combine similar terms. Add the terms with \(m\): \(12m + m = 13m\). Combine the constant numbers: \(132 - 11 = 121\).The final, simplified expression is \(13m + 121\).
By following these steps, you'll be able to simplify even more complex algebraic expressions. Simplification makes expressions easier to work with and is crucial in solving algebraic equations effectively.