Solving for the Number: A Comprehensive Guide

Algebra is an essential branch of mathematics that forms the foundation for advanced mathematical concepts and problem-solving. This guide aims to help students and learners understand the process of solving complex equations through a series of examples. We will walk through the methods and techniques needed to solve various types of algebraic equations, starting from basic linear equations to more complex ones.

Example 1: Basic Equation

Let the number be N. We are given the equation 8N - 1 15. To solve for N, we follow these steps:

Add 1 to both sides of the equation to isolate the term with the variable:

8N - 1 1 15 1

8N 16

Divide both sides of the equation by 8 to solve for N:

N 16 ÷ 8 2

Example 2: Understanding Fractions

Consider the equation 7 * N 15 ÷ 2. Here, we need to understand fractions and how to manipulate them. The steps are as follows:

Translate the division into a fraction:

N (15 / 2) 7 1/2 or 7.5

To break it down further, we can express 15 as a fraction:

15 / 2 (15 * 1 / 2 * 1) 15 / 2

Which simplifies to 7 1/2 or 7.5

Example 3: Solving for a Fraction

Let's consider the equation 1 - 8m 15. Here, we need to solve for m:

Isolate the term with m by subtracting 1 from both sides of the equation:

1 - 1 - 8m 15 - 1

-8m 14

Divide both sides by -8 to solve for m:

m 14 ÷ -8 -7/4

Example 4: More Complex Equations

Let the number be number, where number 2a. The equation given is number - 9 15. Here's how to solve for a:

Replace number with 2a in the equation:

2a - 9 15

Add 9 to both sides to isolate the term with a:

2a - 9 9 15 9

2a 24

Divide both sides by 2 to solve for a:

a 24 ÷ 2 12

Conclusion

These examples demonstrate the step-by-step process of solving various algebraic equations. Whether it's a simple linear equation or a more complex one involving fractions, understanding the fundamental principles is key. By practicing these methods, you can enhance your problem-solving skills and gain confidence in handling algebraic expressions.