Problem 72 - Algebra A Combined Function

Question

Simplify each expression. See Section 1.8. $$ -3(w+7)+5(w+1) $$

Step-by-Step Solution

Answer

2w - 16.

Step 1: Distribute the constants

First, we need to distribute the constants outside the parentheses into the terms inside the parentheses. For the expression \[-3(w+7)+5(w+1),\]distribute -3 to both \(w\) and \(7\), and 5 to both \(w\) and \(1\): -3(w) + (-3)(7) + 5(w) + 5(1).

Step 2: Simplify the distributed expression

Now simplify each distributed expression: \(-3w - 21 + 5w + 5\).

Step 3: Combine like terms

Combine the like terms in the simplified expression. Here, the like terms are the terms with \(w\), and the constant terms:\((-3w + 5w) + (-21 + 5)\).This results in:\(2w - 16\).

Key Concepts

Understanding the Distributive Property

The distributive property in algebra is a rule that helps us to break down expressions and make calculations easier. It involves "distributing" a number that is outside parentheses into each term inside the parentheses. This property is crucial when simplifying algebraic expressions, especially when dealing with variables.

Imagine we have an expression like rr\[-3(w+7)+5(w+1)\].

By applying the distributive property, we multiply the numbers outside each set of parentheses by each term inside the parentheses. That means:

• Multiply
  \(-3\) by \(w\) and \(7\), yielding \(-3w - 21\).


• Multiply \(5\) by \(w\) and \(1\), resulting in \(5w + 5\).

Through these steps, we've expanded the expression and removed the parentheses, setting us up for the next stages of simplification.

Combining Like Terms

Once we have applied the distributive property, the expression typically contains several terms, some of which are "like terms." Like terms are terms that have the exact same variable raised to the same power or no variable at all.

In our expanded expression rr\(-3w - 21 + 5w + 5\),

we identify like terms:

• \(-3w\) and \(5w\) are like terms because they both contain the variable \(w\).
• \(-21\) and \(5\) are like terms because they are both constant terms without variables.

To combine like terms, add their coefficients. Coefficients are the numbers in front of the variables:

• Combine \(-3w + 5w\) which results in \(2w\).
• Combine \(-21 + 5\) giving \(-16\).

Combining like terms simplifies the expression further and makes it easier to interpret or solve.

Simplifying Expressions

The process of simplifying algebraic expressions involves applying several algebraic rules to reduce the expression to its simplest form. Simplifying makes expressions easier to use in equations and presents them in a cleaned-up way.

In our example, after distributing and combining like terms, we were left with \(2w - 16\).
This is the simplified form of the original expression \[-3(w+7)+5(w+1)\].
To achieve this, we:

• Applied the distributive property to eliminate parentheses.
• Identified and combined like terms.

Through these steps, the expression is now reduced to the simplest form where no further simplification is possible. Simplifying expressions is an essential skill in algebra, ensuring expressions are manageable and equations solvable.
Regular practice helps in mastering this skill, making mathematical problems less complex and more approachable.