Problem 78 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$6 n-2=40$$

Step-by-Step Solution

Answer

The solution is \(n = 7\).

Step 1: Understand the Equation

The equation given is \(6n - 2 = 40\). Our goal is to solve for the variable \(n\). To do this, we need to isolate \(n\) on one side of the equation.

Step 2: Add 2 to Both Sides

To begin isolating \(n\), first eliminate the constant term on the left side by adding 2 to both sides of the equation. This will give us \(6n - 2 + 2 = 40 + 2\). Simplifying both sides results in \(6n = 42\).

Step 3: Divide Both Sides by 6

Now that the equation is \(6n = 42\), divide both sides by 6 to solve for \(n\). Doing this yields \(n = \frac{42}{6}\).

Step 4: Simplify the Division

Simplify \(\frac{42}{6}\) by performing the division. This gives \(n = 7\). Hence, \(n = 7\) is the solution.

Key Concepts

Isolation of Variables

When solving equations, our main goal is to find the value of the unknown variable. This means we need to have the variable alone on one side of the equation. Think about moving the variable to a quieter place, free from other numbers. When we have an equation like \(6n - 2 = 40\), the variable \(n\) is mixed with other numbers. To isolate it, we perform arithmetic operations that eliminate these extra numbers from its side. In our example, we remove the \(-2\) by adding 2 to both sides. This step-by-step removal of unwanted numbers is what we call isolating the variable. Once you master this concept, solving equations becomes much easier and simpler!

Linear Equations

A linear equation is one of the most basic forms of algebraic equations. It can often be spotted because it has variables, like \(n\), raised only to the first power. In our example equation \(6n - 2 = 40\), you only see \(n\) without any exponents (like squares or cubes), which makes it linear. Linear equations graph as straight lines when plotted on a coordinate plane. They're the stepping stones of algebra because they're straightforward and easy to manipulate. Understanding how to work with them lays the groundwork for dealing with more complicated equations later on. Remember, linear equations are all about balance—whatever you do to one side, you do to the other to keep this balance in place.

Simplifying Expressions

Simplifying expressions is about making math as neat and tidy as possible. When we simplify, we're looking to reduce expressions to their simplest form. Take the equation step from our exercise, \(6n - 2 + 2 = 40 + 2\). By simplifying each part, we combined like terms to get \(6n = 42\). The simplification didn't stop there. In the next step, we divided 42 by 6, and got \(n = 7\). Simplification helps by stripping away the clutter, leaving only the core parts we need to solve the equation. Think of it as tidying up a room—you're keeping only what's necessary. This process is crucial in making equations manageable and solutions clear.