Use The General Solution To Solve 5 6x 8x 17

Solving 5 + 6x = 8x + 17: A Comprehensive Guide to Linear Equations

This article provides a step-by-step guide on how to solve the linear equation 5 + 6x = 8x + 17, explaining the underlying principles and demonstrating the application of the general solution method. We'll delve into the concepts of linear equations, variables, constants, and the process of isolating the variable to find its value. Understanding this process is fundamental to mastering algebra and solving a wide range of mathematical problems.

Understanding Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable raised to the power of 1. The general form of a linear equation with one variable is:

ax + b = c

where:

• a is the coefficient of the variable x (a number multiplied by x).
• x is the variable we need to solve for.
• b is a constant term on the left side of the equation.
• c is a constant term on the right side of the equation.

Our example equation, 5 + 6x = 8x + 17, fits this form. Here, a = -2 (explained below), b = 5, and c = 17.

Solving the Equation: 5 + 6x = 8x + 17

Our goal is to isolate the variable 'x' to find its value. We achieve this by performing a series of algebraic manipulations, ensuring that we maintain the equality of both sides of the equation. Each step should be clearly documented for accuracy and to aid understanding.

Step 1: Combine Like Terms

The first step involves simplifying the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have terms with 'x' and constant terms. Let's group them together:

To do this effectively, we need to move all terms containing 'x' to one side of the equation and all constant terms to the other side. We can achieve this by adding or subtracting the same value from both sides.

Step 2: Move Variable Terms to One Side

Let's subtract 6x from both sides of the equation:

5 + 6x - 6x = 8x + 17 - 6x

This simplifies to:

5 = 2x + 17

Step 3: Move Constant Terms to the Other Side

Now, let's move the constant term 17 to the left side by subtracting 17 from both sides:

5 - 17 = 2x + 17 - 17

This simplifies to:

-12 = 2x

Step 4: Isolate the Variable

We have 2x = -12. To isolate 'x', we need to divide both sides of the equation by the coefficient of x, which is 2:

-12 / 2 = 2x / 2

This gives us the solution:

x = -6

Verifying the Solution

It's crucial to verify our solution by substituting the value of x back into the original equation:

5 + 6(-6) = 8(-6) + 17

5 - 36 = -48 + 17

-31 = -31

Since the equation holds true, our solution x = -6 is correct.

General Solution and its Applications

The method used above is a general approach for solving linear equations. This method is applicable to various real-world problems that can be modeled using linear equations. These problems range from simple calculations involving distances, speeds, and time to more complex scenarios involving financial modeling, physics, and engineering.

For instance, consider a problem where you need to determine the number of hours you need to work at a certain hourly rate to earn a specific amount of money after deducting taxes. This can easily be represented by a linear equation where the unknown is the number of hours worked. Solving this equation using the general approach described above allows you to determine the required number of working hours.

Expanding on the Concept: Simultaneous Equations

While we focused on a single linear equation, the principles extend to solving systems of simultaneous linear equations. These systems involve two or more linear equations with two or more variables. Methods like substitution, elimination, and matrix methods are used to solve these systems. Understanding the fundamental principles of solving single linear equations is essential for tackling these more complex scenarios. Each method has its own advantages and disadvantages, and choosing the best approach often depends on the specific structure of the equations.

Advanced Applications in Calculus and Beyond

The concepts of linear equations and their solutions form the foundation for more advanced mathematical topics. In calculus, linear approximations are used to estimate the behavior of functions near specific points. Linear equations also play a critical role in differential equations, a branch of mathematics that deals with equations involving derivatives. These applications extend into various fields including physics, engineering, and economics where linear models are often used to simplify complex phenomena.

Troubleshooting Common Mistakes

Many students encounter common mistakes when solving linear equations. Understanding these common errors and knowing how to avoid them is vital for mastering the concept.

• Incorrect Sign Handling: A common error is making mistakes with positive and negative signs when adding, subtracting, multiplying, or dividing. Double-checking each step carefully helps prevent these errors.

• Incorrect Order of Operations: Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Failure to follow this order can lead to incorrect solutions.

• Errors in Fraction Manipulation: When dealing with fractions, ensure you correctly apply the rules of fraction addition, subtraction, multiplication, and division. Simplify fractions whenever possible to reduce the complexity of the equation.

• Not Checking the Solution: Always verify your solution by substituting the value of the variable back into the original equation. This simple step helps catch errors and ensures accuracy.

Conclusion

Solving the linear equation 5 + 6x = 8x + 17, as demonstrated, involves a systematic approach of combining like terms, isolating the variable, and verifying the solution. This seemingly simple equation illustrates the fundamental principles underlying the solution of linear equations, principles that extend to more complex mathematical problems and real-world applications. Mastering this process is not only crucial for success in algebra but also builds a solid foundation for future mathematical studies and problem-solving skills across diverse fields. Careful attention to detail, consistent practice, and understanding the underlying concepts will lead to success in solving linear equations and other algebraic problems. Remember to always check your work to ensure accuracy and confidence in your solutions.