Solving Linear Equations: A Comprehensive Guide with Examples

Linear equations are fundamental in algebra and form the basis for more complex mathematical concepts. They can be found in various real-world applications, from finance to physics. In this guide, we will explore how to solve a linear equation using a specific example and provide step-by-step instructions. We’ll also delve into the General Form of a Linear Equation and its significance in understanding the behavior of the equation.

Solving the Linear Equation 2/3x - 1/5x 7

Let's consider the linear equation: 2/3x - 1/5x 7. This equation might look intimidating at first, but with a systematic approach, it becomes quite straightforward to solve.

Step 1: Convert the Equation to a Common Denominator

The first step involves converting the equation to a common denominator. The least common denominator (LCD) for 3 and 5 is 15. We can rewrite the equation as:

(2/3)x - (1/5)x 7 becomes (10/15)x - (3/15)x 7

Step 2: Simplify the Equation

Now that we have a common denominator, we can combine the terms on the left side of the equation:

(10/15)x - (3/15)x 7

Step 3: Perform the Algebraic Operation

Next, we simplify the fraction:

(10/15 - 3/15)x 7

Step 4: Solve for x

The simplified equation is:

(7/15)x 7

To isolate x, multiply both sides of the equation by the reciprocal of 7/15:

x 7 * (15/7) 15

Verification

To verify our solution, we substitute x 15 back into the original equation:

(2/3)(15) - (1/5)(15) 7

10 - 3 7

Thus, x 15 is indeed the solution to the equation.

General Form of a Linear Equation

The general form of a linear equation is ax b 0, where a and b are constants. In the given example, the equation can be rewritten in this form to understand its structure better. The equation 2/3x - 1/5x 7 can be written as:

(2/3 - 1/5)x 7

This can be simplified to:

(10/15 - 3/15)x 7

(7/15)x 7

Conclusion

Understanding how to solve linear equations is crucial in many fields of study. By following a structured approach and verifying the solution, we can confidently solve linear equations. In this article, we have provided a detailed step-by-step guide to solving the equation 2/3x - 1/5x 7, and we have also discussed the general form of a linear equation. Whether you are a student, a teacher, or a professional, a strong grasp of these concepts is invaluable.

Additional Resources

For further reading, you might want to explore additional resources on linear equations and algebraic solutions. Websites like Khan Academy and Coursera offer comprehensive courses that delve into these topics in greater detail. Books such as "Algebra: Structure and Method" by Richard G. Brown and "Elementary Algebra" by Lynn Marecek and Mary Ann Connors can also be very helpful.