Solving 5x - 1 = 64 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. It allows us to unravel the mysteries behind unknown quantities and find solutions that satisfy specific conditions. Linear equations, in particular, form the bedrock of algebraic problem-solving. They are characterized by a single variable raised to the power of one, and they represent a direct relationship between the variable and a constant term. In this article, we will delve into the step-by-step process of solving a linear equation, using the example of 5x - 1 = 64. This comprehensive guide will equip you with the knowledge and techniques to tackle similar equations with confidence.

Understanding the Basics of Linear Equations

Before we embark on the journey of solving our equation, let's take a moment to grasp the fundamental concepts behind linear equations. A linear equation is an algebraic statement that asserts the equality of two expressions, where the variable appears only to the first power. The general form of a linear equation is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' represents the variable we aim to find.

In essence, solving a linear equation involves isolating the variable on one side of the equation. This means manipulating the equation through algebraic operations until we have the variable standing alone, with its value explicitly revealed on the other side. The key to this process lies in maintaining the equality of both sides throughout the transformations.

Step 1: Isolating the Variable Term

Our equation, 5x - 1 = 64, presents us with a variable term (5x) and a constant term (-1) on the left-hand side. To isolate the variable term, we need to eliminate the constant term. This is achieved by performing the inverse operation of subtraction, which is addition. We add 1 to both sides of the equation, ensuring that the balance remains intact:

5x - 1 + 1 = 64 + 1

This simplifies to:

5x = 65

Now, we have successfully isolated the variable term (5x) on the left-hand side, paving the way for the next step.

Step 2: Solving for the Variable

The variable 'x' is currently multiplied by the coefficient 5. To isolate 'x' completely, we need to undo this multiplication. The inverse operation of multiplication is division. We divide both sides of the equation by 5:

5x / 5 = 65 / 5

This simplifies to:

x = 13

And there we have it! We have successfully solved the equation and found the value of 'x' to be 13. This means that when we substitute 13 for 'x' in the original equation (5x - 1 = 64), the equation holds true.

Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it. We substitute x = 13 back into the original equation:

5 * 13 - 1 = 64

65 - 1 = 64

64 = 64

The equation holds true, confirming that our solution, x = 13, is indeed correct.

Presenting the Solution

The problem statement instructed us to express the solution as a reduced fraction or an integer. In this case, our solution, 13, is already an integer, so we don't need to perform any further simplification.

Alternative Scenarios: No Real Solution or Multiple Solutions

While our equation had a single, unique solution, it's important to be aware of scenarios where linear equations may have no real solutions or multiple solutions.

No Real Solution

An equation may have no real solution if it leads to a contradiction. For example, consider the equation:

2x + 5 = 2x - 3

If we attempt to solve this equation, we would subtract 2x from both sides, resulting in:

5 = -3

This statement is clearly false, indicating that there is no value of 'x' that can satisfy the equation. In such cases, we would write "DNE" (Does Not Exist) as the solution.

Multiple Solutions

Linear equations can also have infinitely many solutions if they are identities. An identity is an equation that is true for all values of the variable. For example, consider the equation:

3x + 6 = 3(x + 2)

If we simplify the right-hand side, we get:

3x + 6 = 3x + 6

This equation is true for any value of 'x'. In such cases, we would indicate that there are infinitely many solutions.

Conclusion: Mastering the Art of Solving Linear Equations

Solving linear equations is a fundamental skill in mathematics, laying the groundwork for more advanced algebraic concepts. By understanding the principles of isolating the variable and maintaining the equation's balance, you can confidently tackle a wide range of linear equations. Remember to always verify your solution to ensure accuracy and be mindful of scenarios where no real solutions or multiple solutions may exist. With practice and perseverance, you can master the art of solving linear equations and unlock the doors to further mathematical exploration.

In this article, we meticulously dissected the process of solving the linear equation 5x - 1 = 64. We started by understanding the basics of linear equations, then proceeded with the step-by-step solution, which involved isolating the variable term and solving for the variable. We also emphasized the importance of verifying the solution to ensure its accuracy. Furthermore, we explored alternative scenarios where linear equations may have no real solutions or multiple solutions. By grasping these concepts and techniques, you will be well-equipped to conquer any linear equation that comes your way.

Solving linear equations is a core skill in algebra. To effectively tackle these equations, remember these key takeaways:

• Isolate the variable term: Use inverse operations (addition/subtraction) to get the term containing 'x' alone on one side of the equation.
• Solve for the variable: Use inverse operations (multiplication/division) to get 'x' by itself.
• Verify your solution: Substitute the value you found for 'x' back into the original equation to make sure it holds true.
• Be aware of special cases: Some equations have no solution (DNE), and some have infinitely many solutions.

To solidify your understanding, try solving these practice problems:

1. Solve for x: 2x + 3 = 7
2. Solve for y: 4y - 5 = 11
3. Solve for z: 6z + 2 = 2z - 6
4. Solve for a: 7a - 4 = 3a + 8

By working through these problems, you'll gain confidence and proficiency in solving linear equations. Remember, practice makes perfect! Embrace the challenges, and you'll become a master of algebra in no time.

Mastering the art of solving linear equations is a vital step in your mathematical journey. By understanding the fundamental principles and practicing regularly, you'll develop the skills and confidence to tackle more complex problems in the future. So, keep exploring, keep learning, and keep solving!