Problem 11 - Prealgebra

Question

Solve equation. Check your solution. \(n-14=3 n\)

Step-by-Step Solution

Answer

The solution is \(n = -7\).

Step 1: Move Variables to One Side

In this step, we need to move the variable terms to one side of the equation. Subtract \(n\) from both sides to eliminate \(n\) on the left side. \[ n - 14 - n = 3n - n \] This simplifies to: \[ -14 = 2n \]

Step 2: Solve for the Variable

We can now solve for \(n\). Divide both sides by 2 to isolate \(n\).\[\frac{-14}{2} = n \]This simplifies to: \[ n = -7 \]

Step 3: Check the Solution

Substitute \(n = -7\) back into the original equation to ensure it satisfies the equation. Starting with the left side: \(n - 14\) = \(-7 - 14 = -21\).Now the right side: \(3n = 3(-7) = -21\). Since both sides are equal, \(n = -7\) is a valid solution.

Key Concepts

Understanding Prealgebra Concepts

Prealgebra is the foundation of your algebra journey. It's important because it prepares you for more complex equations by introducing root concepts. At its core, prealgebra focuses on helping you understand the basic properties of numbers and operations. This includes, but is not limited to:

• Addition, subtraction, multiplication, and division — the very building blocks of all math.
• Understanding variables — symbols like \( n \) that stand for unknown values.
• Setting up and simplifying expressions — combining terms and reducing equations to simpler forms.

Grasping these fundamentals equips you with the tools necessary to tackle equations confidently.

Mastering Variable Isolation

Variable isolation is a key concept that helps solve equations. The goal is to get the variable, such as \( n \), by itself on one side of the equation. Let's consider our original equation:\[ n - 14 = 3n \]Initially, you will want to eliminate extra terms involving the variable by moving them to one side of the equation. This often involves:

• Addition or subtraction to remove terms.
• Division or multiplication to simplify factors.

For example, in the step-by-step solution, we subtract \( n \) from both sides, resulting in:\[ -14 = 2n \]This leaves us with our variable part, \( 2n \), on one side, simplifying the problem.

Checking Solutions

Checking solutions is crucial to ensure your answer is correct. Once you've found a potential solution, like \( n = -7 \), you plug it back into the original equation to confirm it balances. This verification involves:

• Substituting the value back into both sides of the equation.
• Simplifying both sides separately to see if they equal.

In our example, substituting \( n = -7 \) gives us both sides of the equation as \(-21\), confirming the correctness. If both sides match, your solution is verified. If not, double-check the calculation process for errors.

Exploring Basic Arithmetic Operations

Basic arithmetic operations are at the heart of solving algebraic equations. These include addition, subtraction, multiplication, and division. Each operation has a specific role in manipulating equations:

• Addition and subtraction help in moving terms across the equation.
• Multiplication and division come into play when simplifying equations to isolate variables.

For instance, in the problem, after isolating the term \(2n\), we divided both sides by 2 (a division operation) to solve for \( n \):\[ \frac{-14}{2} = n \]This basic understanding of arithmetic operations is essential, as they form the basis of clear, logical solutions.