Problem 43 - Algebra 2

Question

Simplify each expression. $$ 7(0.2 p+0.3 q)+5(0.6 p-q) $$

Step-by-Step Solution

Answer

The simplified expression is \( 4.4p - 2.9q \).

Step 1: Distribute within each term

First, apply the distributive property to each parenthetical expression.1. Distribute the 7 into the first group: \( 7(0.2p + 0.3q) = 7 \times 0.2p + 7 \times 0.3q = 1.4p + 2.1q \) 2. Distribute the 5 into the second group: \( 5(0.6p - q) = 5 \times 0.6p + 5 \times -1q = 3.0p - 5q \) Now, the expression becomes:\( 1.4p + 2.1q + 3.0p - 5q \)

Step 2: Combine like terms

Look for terms with the same variable. The expression has terms with \(p\) and terms with \(q\). 1. Combine \(p\) terms: \( 1.4p + 3.0p = 4.4p \) 2. Combine \(q\) terms: \( 2.1q - 5q = -2.9q \) This simplifies the expression to:\( 4.4p - 2.9q \)

Key Concepts

Distributive property

The distributive property is a fundamental tool in algebra that helps us expand expressions. Think of it as a way to "distribute" multiplication over addition or subtraction within the parentheses. This property is particularly useful when you're dealing with expressions that include parentheses, as it allows you to eliminate them and simplify your expression step by step.

For example, in the expression \( 7(0.2p + 0.3q) \), the distributive property allows us to multiply each term inside the parenthesis by 7. This gives us \( 7 \times 0.2p + 7 \times 0.3q \), which simplifies to \( 1.4p + 2.1q \).

Here's a quick way to remember why it's called the "distributive" property:

• You "distribute" the number outside the parenthesis to each part inside.
• It works for both addition and subtraction.

This method not only simplifies your work but also sets the stage for the next steps in simplifying expressions.

Combining like terms

Combining like terms is the next important step in simplifying algebraic expressions. Once you've used the distributive property, you'll often end up with multiple terms that contain the same variables. By combining these, you further simplify the expression.

In the expression \( 1.4p + 2.1q + 3.0p - 5q \), notice that there are two terms with the variable \( p \) and two terms with the variable \( q \). To combine like terms, simply add or subtract the coefficients of the terms that contain the same variable.

Here’s how to do it:

• For terms with \( p \): Combine \( 1.4p \) and \( 3.0p \) to get \( 4.4p \).
• For terms with \( q \): Combine \( 2.1q \) and \( -5q \) to get \( -2.9q \).

By organizing and combining these like terms, you ensure the expression is as simple as possible, making it easier to evaluate or use in further problem-solving.

Simplifying expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to use in further calculations or evaluations. After using the distributive property and combining like terms, you're often left with a much cleaner expression. This process is crucial to help clarify complex algebraic expressions and make them manageable.

The result from our exercise is expressed as \( 4.4p - 2.9q \). This is the simplified form because it has no parentheses, and no like terms can be further combined. When you simplify expressions, you are essentially peeling away layers of complexity to reveal an expression that's straightforward and concise.

Why is this step necessary?

• It makes calculations easier and more straightforward.
• It helps in recognizing relationships between variables.
• It prepares expressions for solving equations or inequalities.

Understanding how to simplify expressions helps develop strong algebraic skills, which are foundational for advanced math topics.