Problem 12 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(8+3(s-6 t)\)

Step-by-Step Solution

Answer

8 + 3s - 18t

Step 1: Distribute the 3

Distribute the coefficient 3 inside the parenthesis to both terms inside the parenthesis. This means you will multiply 3 by both \(s\) and \(-6 t\).

Step 2: Perform the multiplication

Multiply 3 by \(s\) to get \(3s\), and multiply 3 by \(-6 t\) to get \(-18t\). This gives us the expression: \(8 + 3s - 18t\).

Step 3: Combine like terms

There are no like terms to combine here. So the simplified expression remains: \(8 + 3s - 18t\).

Key Concepts

Distribution in Algebra

In algebra, distribution refers to the process of expanding expressions that involve parentheses. This is done by multiplying a term outside the parenthesis by each term inside the parenthesis. For example, in the expression \(3(s - 6t)\), the number 3 is multiplied with both \(s\) and \(-6t\). Specifically:

* Multiply 3 by \(s\), which gives you \(3s\).
* Multiply 3 by \(-6t\), which results in \(-18t\).

After performing these multiplications, the expression \(3(s - 6t)\) is distributed to become \(3s - 18t\). This step is fundamental in transforming more complex algebraic expressions into simpler forms that can be more easily worked with or combined.

Combining Like Terms

Combining like terms is a key step in simplifying algebraic expressions. Like terms are terms that have identical variable parts, meaning the same variables raised to the same powers. In the simplified expression \(8 + 3s - 18t\), there are no like terms to combine. The constant term 8, the term \(3s\), and the term \(-18t\) are all distinct and cannot be added together. Here's how you identify and combine like terms:

* Look for terms that have the same variable raised to the same exponent.
* Add or subtract the coefficients of these like terms.

If the expression had been \(8 + 3s - 18t + 4s\), you could combine \(3s\) and \(4s\) to get \(7s\), resulting in \(8 + 7s - 18t\). This consolidates the expression into a simpler form.

Algebraic Multiplication

Algebraic multiplication involves multiplying numbers and variables according to the distributive property. In the expression \(3(s - 6t)\), the multiplication is carried out term by term. Here’s the detailed process:

* Multiply the coefficient outside the parenthesis (3) by the first term inside the parenthesis (\(s\)). This gives you \(3s\).
* Then, multiply the same coefficient (3) by the second term inside the parenthesis (\(-6t\)). This results in \(-18t\).

This process, known as distribution, ensures every term inside the parenthesis is multiplied by the factor outside it. This leads to the expression being expanded and simplified into a format where further algebraic operations, such as combining like terms, can be executed more straightforwardly. Keeping track of both coefficients and variable terms during multiplication is essential for maintaining accuracy in algebraic simplification.