Problem 62 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$m(5 m-2)+9(5-m)$$

Step-by-Step Solution

Answer

The simplified expression is \( 5m^2 - 11m + 45 \).

Step 1: Distribute in the first term

Multiply the terms inside the first parentheses by the factor in front of it. The expression is \( m(5m-2) \). Distribute the \( m \) to get \( 5m^2 - 2m \).

Step 2: Distribute in the second term

Multiply the terms inside the second parentheses by the factor in front of it. The expression is \( 9(5-m) \). Distribute the \( 9 \) to get \( 45 - 9m \).

Step 3: Combine like terms

Now combine all terms from both distributed parts of the expression: \( 5m^2 - 2m + 45 - 9m \). Combine the \(-2m\) and \(-9m\) to get \(-11m\).

Step 4: Write simplified expression

The final expression is what you get after simplifying: \( 5m^2 - 11m + 45 \).

Key Concepts

Distributive Property

The distributive property is a fundamental principle of algebra that makes it easier to handle expressions within parentheses. This involves multiplying a single term by each term inside a set of parentheses. In the original exercise, you are given expressions such as \( m(5m-2) \) and \( 9(5-m) \). Here’s how the distributive property works:

• First, multiply the term outside the parentheses by each term inside the parentheses.
• This means multiplying \( m \) by both \( 5m \) and \(-2\) separately to get \( 5m^2 - 2m \).
• Similarly, multiply \( 9 \) by each term in \( 5-m \), resulting in \( 45 - 9m \).

Use the distributive property whenever you need to "distribute" the outer term to each term in the parenthetical expression.
This process helps in transforming a complex expression into a more workable form, easing further calculations.

Combining Like Terms

Combining like terms is the action of simplifying expressions by merging terms that have the same variables raised to the same power. This is important because it reduces the complexity of an algebraic expression, making it easier to solve or further simplify.
For example, after applying the distributive property, the expression becomes \(5m^2 - 2m + 45 - 9m\). You can spot the like terms by looking for terms that have the same variable component. In this case:

• \(-2m\) and \(-9m\) are like terms because they both contain \(m\)

Add these terms together by adding their coefficients to get \(-11m\).
This concept is foundational in obtaining a cleaner and more concise form of an algebraic expression, making it easier to visualize and solve.

Simplifying Expressions

Simplifying expressions is a process that combines all the steps necessary to transform a complex algebraic expression into its most basic form. It involves using properties like the distributive property and combining like terms.
In the original problem, after distributing and combining like terms, you arrive at the expression \( 5m^2 - 11m + 45 \). Simplification ensures clarity and facilitates easier interpretation or further algebraic procedures.

• Simplifying makes equations easier to work with by reducing unnecessary complexity.
• It also helps in easily identifying patterns or errors that might have been overlooked.

This simplified form is often the one used in equations, making calculations and problem-solving more efficient. Understanding how to simplify is key to learning algebra effectively and paves the way for tackling more complex problems in mathematics.