Solving 5x - 1 = 2x - 13 A Step-by-Step Guide

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Introduction

In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, serve as the bedrock for more advanced mathematical concepts. This article delves into the step-by-step process of solving the linear equation 5x - 1 = 2x - 13, providing a comprehensive guide suitable for students and anyone looking to refresh their algebra skills. We'll break down each step, explain the underlying principles, and offer additional insights to solidify your understanding. This guide aims to empower you with the knowledge and confidence to tackle similar problems with ease. Understanding how to manipulate equations and isolate variables is not just about getting the right answer; it's about developing a logical and systematic approach to problem-solving, a skill that extends far beyond the classroom.

Understanding Linear Equations

Before diving into the solution, it's crucial to understand what a linear equation is. 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. The variable's highest power is one. In simpler terms, when graphed on a coordinate plane, a linear equation forms a straight line. Our equation, 5x - 1 = 2x - 13, fits this definition perfectly. It involves a single variable, x, and the highest power of x is one. Linear equations are ubiquitous in mathematics and its applications, appearing in physics, engineering, economics, and computer science. They model various real-world phenomena, from simple relationships like distance and speed to more complex systems. Mastering the art of solving linear equations is therefore an essential step in your mathematical journey.

Core Principles

The core principle behind solving any equation, including linear equations, is maintaining balance. Imagine an equation as a weighing scale; both sides must remain equal to keep the scale balanced. Any operation performed on one side of the equation must also be performed on the other side to preserve this equality. This principle is based on the fundamental properties of equality in algebra: the addition property, the subtraction property, the multiplication property, and the division property. These properties state that adding, subtracting, multiplying, or dividing both sides of an equation by the same number (except for division by zero) does not change the equation's solution. Understanding and applying these properties are key to isolating the variable and finding its value. In the following sections, we'll see how these properties are used step-by-step to solve our equation.

Step-by-Step Solution of 5x - 1 = 2x - 13

Step 1: Grouping Like Terms

The first step in solving 5x - 1 = 2x - 13 is to group like terms together. Like terms are terms that contain the same variable raised to the same power (or are constants). In our equation, the terms with x are 5x and 2x, and the constant terms are -1 and -13. To group these terms, we want to move all the x terms to one side of the equation and all the constant terms to the other side. A common strategy is to move terms to the side where the variable term has the larger coefficient. In this case, 5x has a larger coefficient than 2x, so we'll move the 2x term to the left side of the equation. To do this, we subtract 2x from both sides of the equation:

5x - 1 - 2x = 2x - 13 - 2x

This simplifies to:

3x - 1 = -13

Now, we need to move the constant term -1 to the right side of the equation. To do this, we add 1 to both sides:

3x - 1 + 1 = -13 + 1

This simplifies to:

3x = -12

At this point, we've successfully grouped the like terms, isolating the variable term 3x on the left side and the constant term -12 on the right side. This simplification brings us closer to our goal of solving for x.

Step 2: Isolating the Variable

After grouping like terms, the next step is to isolate the variable x. In our simplified equation, 3x = -12, x is being multiplied by 3. To isolate x, we need to undo this multiplication. The inverse operation of multiplication is division, so we divide both sides of the equation by the coefficient of x, which is 3:

(3x) / 3 = (-12) / 3

This simplifies to:

x = -4

By dividing both sides by 3, we've successfully isolated x and found its value. The equation now clearly shows that x is equal to -4. This step demonstrates the power of using inverse operations to undo mathematical operations and isolate variables. Isolating the variable is a critical skill in algebra, and mastering it allows you to solve a wide range of equations.

Step 3: Verifying the Solution

After finding a solution, it's always a good practice to verify it. This ensures that the solution is correct and that no errors were made during the solving process. To verify our solution, we substitute the value of x we found (x = -4) back into the original equation, 5x - 1 = 2x - 13. If the left side of the equation equals the right side after the substitution, our solution is correct. Let's substitute x = -4 into the equation:

5(-4) - 1 = 2(-4) - 13

Now, we simplify both sides:

-20 - 1 = -8 - 13

-21 = -21

Since both sides of the equation are equal, our solution x = -4 is correct. This verification step not only confirms our answer but also reinforces our understanding of the equation and the solving process. Verifying solutions is a valuable habit to develop in mathematics, as it helps catch mistakes and build confidence in your problem-solving abilities.

Common Mistakes and How to Avoid Them

Solving equations can be tricky, and it's easy to make mistakes along the way. Here are some common mistakes students make when solving linear equations, along with tips on how to avoid them:

  1. Incorrectly Applying the Distributive Property: The distributive property states that a(b + c) = ab + ac. A common mistake is forgetting to distribute the 'a' to both 'b' and 'c'. Always ensure you multiply the term outside the parentheses by every term inside the parentheses.
  2. Combining Unlike Terms: Only like terms (terms with the same variable and exponent) can be combined. For example, 3x and 2x can be combined, but 3x and 2x² cannot. Make sure you are only adding or subtracting terms that have the same variable and exponent.
  3. Forgetting to Perform Operations on Both Sides: The golden rule of equation solving is that any operation performed on one side must also be performed on the other side. Forgetting this leads to an unbalanced equation and an incorrect solution. Always remember to apply the same operation to both sides of the equation to maintain balance.
  4. Sign Errors: Mistakes with negative signs are very common. Pay close attention to the signs of the terms when moving them across the equals sign or when performing operations. Double-check your signs at each step to minimize errors.
  5. Dividing by Zero: Division by zero is undefined. If you encounter a situation where you need to divide by zero, it indicates an error in the equation or the solving process. Always be mindful of division by zero and re-evaluate your steps if it occurs.

By being aware of these common mistakes and taking the time to check your work, you can significantly improve your accuracy in solving linear equations.

Practice Problems

To solidify your understanding of solving linear equations, here are a few practice problems. Try solving them on your own, following the steps outlined in this guide:

  1. 3x + 5 = 14
  2. 2(x - 1) = 4x + 6
  3. 7x - 3 = 5x + 9
  4. -4x + 8 = 2x - 10
  5. 6x + 2 = -4x - 18

Working through these problems will help you gain confidence and proficiency in solving linear equations. Remember to check your answers by substituting them back into the original equations.

Conclusion

Solving linear equations is a fundamental skill in mathematics. By understanding the principles of equality, grouping like terms, isolating variables, and verifying solutions, you can confidently tackle a wide range of algebraic problems. This article provided a detailed step-by-step solution to the equation 5x - 1 = 2x - 13, along with insights into common mistakes and practice problems to further enhance your understanding. Remember that practice is key to mastering this skill. Keep solving equations, and you'll find yourself becoming more proficient and confident in your mathematical abilities. The ability to solve equations is not just a mathematical skill; it's a powerful tool for problem-solving in various aspects of life. Embrace the challenge, and you'll unlock a world of mathematical possibilities.