Problem 371 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ (5 m-3)-(m+7) $$

Step-by-Step Solution

Answer

4m - 10

Step 1: Identify the expression

The expression to simplify is \( (5m - 3) - (m + 7) \).

Step 2: Distribute the negative sign

Apply the distributive property to remove the parentheses: \( 5m - 3 - m - 7 \).

Step 3: Combine like terms

Combine the terms with \( m \) and the constant terms: \( 5m - m - 3 - 7 \), which simplifies to \( 4m - 10 \).

Key Concepts

simplifying expressions

Simplifying expressions is a key skill in algebra. It means making an expression easier to understand or solve. To simplify an expression, we can use various algebraic techniques like the distributive property, combining like terms, and understanding coefficients. Always start by identifying parts of the expression that can be simplified. For example, in the expression \( (5m - 3) - (m + 7) \), you need to remove parentheses and then combine like terms. Simplifying not only makes calculations easier but also helps in solving equations effectively.

combining like terms

Combining like terms is an important step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. For instance, in the expression \( 5m - m - 3 - 7 \), we can combine \( 5m \) and \( m \). Here’s how you do it:

• Identify like terms: \( 5m \) and \( -m \) are like terms.
• Add or subtract the coefficients: \( 5m - m = 4m \).

Do the same for constant terms: \( -3 - 7 = -10 \). Thus, the simplified expression is \( 4m - 10 \). This step makes the algebraic expression more manageable and sets the stage for solving equations.

algebraic expressions

Algebraic expressions are a combination of variables, numbers, and operations. They do not have an equality sign like equations. They might look complex, but breaking them down makes them easier to understand. The expression \( (5m - 3) - (m + 7) \) includes:

• Constants: Numbers without variables, like \( -3 \) and \( 7 \).
• Variables: Symbols that represent numbers, like \( m \).
• Coefficients: Numbers multiplying the variables, like \( 5 \) in \( 5m \) and \( 1 \) in \( m \).

Understanding these components helps in simplifying and manipulating the expressions to solve algebra problems. Knowing what each part represents makes the calculation steps clearer.

negative sign distribution

Negative sign distribution, or multiplying by \( -1 \), is a vital step in simplifying expressions. This means distributing a negative sign to all terms within the parentheses. For the expression \( (5m - 3) - (m + 7) \), you distribute the negative sign as follows:

• Rewrite the expression: \( 5m - 3 - 1 \cdot (m + 7) \).
• Distribute the \( -1 \): \( 5m - 3 - m - 7 \).

This step removes the parentheses and changes the signs of the terms inside. Always pay attention to the negative signs to avoid mistakes. This step ensures the expression is ready for the next phase of combining like terms. Properly handling negative signs helps keep your work accurate and simplifies the problem-solving process.