Problem 12 - Elementary Algebra

Question

Solve. $$ 7 n=215 $$

Step-by-Step Solution

Answer

The solution is \( n \approx 30.71 \).

Step 1: Identify the equation

We are given the equation \( 7n = 215 \) and need to solve for the variable \( n \).

Step 2: Isolate the variable

To find \( n \), divide both sides of the equation by 7. This will isolate \( n \) on one side of the equation.

Step 3: Perform the division

Calculate \( n = \frac{215}{7} \). When you divide 215 by 7, you get \( n = 30.7142857 \).

Step 4: Present the solution

The solution to the equation \( 7n = 215 \) is \( n \approx 30.71 \) (rounded to two decimal places).

Key Concepts

Understanding Algebra

Algebra is a crucial part of mathematics that deals with symbols and the rules for manipulating those symbols to solve problems. In algebra, numbers and variables are combined in formulas and equations to represent real-world scenarios or mathematical relationships. When we talk about variables, we're referring to symbols, often letters, that represent unknown values. In our exercise, the variable is \( n \) in the equation \( 7n = 215 \). Our goal is to find the value of \( n \) that makes the equation true.
Algebra helps bridge the gap between arithmetic and more complex mathematics. It's essentially a language of its own, used to express general formulas and solve problems systematically. Many areas of mathematics, science, and engineering rely on algebraic concepts. By learning how to solve algebraic equations, you build a foundation for more advanced topics.
Algebra involves understanding how to manipulate equations to isolate variables, which often requires using operations like addition, subtraction, multiplication, or division. As you become more familiar with these processes, solving algebra problems will become easier and more intuitive.

The Art of Solving Equations

Solving equations is about finding the value of an unknown variable that makes the equation true. The process involves determining what operations can be used to untangle the equation step-by-step until the unknown variable stands alone.
For the equation in our example, \( 7n = 215 \), we start by identifying the operation involving the variable, which is multiplication by 7. To solve for \( n \), we need to reverse this operation. The principle of balancing equations states that whatever you do to one side of the equation, you must do to the other to maintain equality.
This leads us to

• Identify the operation (here it's multiplication by 7).
• Reverse the operation (use division) to isolate \( n \) on one side.
• Perform the division on both sides, giving us \( n = \frac{215}{7} \).

After simplifying, we find \( n \approx 30.71 \). This method ensures that we've correctly solved for the variable without altering the equality of the original equation.

Role of Division in Linear Equations

Division is a fundamental mathematical operation that can be used to isolate a variable when solving linear equations. In our problem, we use division to separate \( n \) from the constant it is multiplied by — 7.
When you divide both sides of the equation \( 7n = 215 \) by 7, you simplify the equation to find \( n \). Essentially, division allows you to "undo" the multiplication that originally connects the variable to a number, leaving the variable isolated as \( n = \frac{215}{7} \).
To perform division accurately:

• Divide each term on one side of the equation by the number you're dividing by.
• Whatever operation is applied to one side of the equation must be applied to the other side to keep the equation balanced.
• Simplify the result: In this example, dividing gives you approximately \( n \approx 30.71 \).

Knowing how division interacts with other operations when solving equations is part of mastering linear equation solutions. It is a valuable skill, not just in mathematics but in understanding real-world problems where dividing quantities is needed.