Problem 77 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$5 n+4=-26$$

Step-by-Step Solution

Answer

The solution to the equation is \(n = -6\).

Step 1: Isolate the variable term

The given equation is \(5n + 4 = -26\). First, we need to isolate the term containing the variable \(n\) by removing the constant term on the left side. We can do this by subtracting 4 from both sides of the equation: \(5n + 4 - 4 = -26 - 4\).This simplifies to:\(5n = -30\).

Step 2: Solve for the variable

Now that we have \(5n = -30\), we need to solve for \(n\) by dividing both sides of the equation by 5:\(\frac{5n}{5} = \frac{-30}{5}\).This simplifies to:\(n = -6\).

Key Concepts

Isolating Variables

Isolating the variable is an essential skill when solving linear equations. The goal is to get the variable, like \( n \), on its own on one side of the equation. This lets us easily determine its value by simplifying the rest of the equation. Let's break down this process familiar in our example.
- Start with the equation: \( 5n + 4 = -26 \). You want to isolate \( n \). To do this, look for terms with the variable on one side and constants on the other side of the equation.
- First, remove the constant from the side with the variable. In the example, this means removing 4 from \( 5n + 4 \). You can subtract 4 from both sides: \( 5n + 4 - 4 = -26 - 4 \), simplifying to \( 5n = -30 \).
By isolating the variable term, we set the stage for solving the equation efficiently. The equation simplifies, making it clear how to find the variable's value.

Step-by-Step Solutions

Step-by-step solutions are key in understanding how to tackle a problem in a structured way. By breaking down each step, you gain insight into each component of an equation and why particular operations are performed. Here's how it works with our example step-by-step solution:

• **Identify the Variable and Constants**: Here, you have \( 5n + 4 = -26 \). Your focus is on \( n \), the variable, and \( 4 \) and \(-26\) are constants.
• **Isolate the Variable**: Start by getting all constants away from the \( n \) term. Subtract 4 from each side to simplify the equation to \( 5n = -30 \).
• **Solve the Simple Equation**: Now that \( n \) is the only variable term, divide each side by the coefficient of \( n \), which is 5. This operation results in \( n = -6 \).

Completing each step methodically ensures a thorough understanding and minimizes error, making it easier for students to solve equations independently.

Prealgebra

Prealgebra lays the foundation for understanding algebra by introducing basic concepts and operations. These concepts include working with numbers, understanding variables, and solving simple equations such as the examples above.
In this prealgebra exercise, we deal with a fundamental equation \(5n + 4 = -26\). Prealgebra helps students become comfortable with:

• **Understanding Variables**: Variables represent unknown numbers that we want to solve for. In our example, \( n \) is a variable.
• **Arithmetic Operations**: Here you perform addition, subtraction, multiplication, and division to solve for \( n \).
• **Balanced Equations**: This means that whatever operation you perform on one side of the equation, you must perform on the other side. This keeps the equation balanced and true.

By mastering these prealgebra skills, students build confidence and prepare themselves for more advanced topics in algebra, securing a solid mathematical foundation.