Solving $5(3x+1) = 50$ A Step-by-Step Guide

Jul 12, 2025 by ADMIN 44 views

Solving Linear Equations A Comprehensive Guide to Mastering $5(3x+1) = 50$

Linear equations are the foundation of algebra, and mastering their solution is crucial for success in mathematics and various real-world applications. This comprehensive guide delves into the step-by-step process of solving the linear equation $5(3x+1) = 50$ , providing a clear understanding of the underlying principles and techniques involved. We will break down each step, explain the rationale behind it, and offer additional insights to enhance your problem-solving skills. Whether you're a student grappling with algebra concepts or simply seeking a refresher, this guide will equip you with the knowledge and confidence to tackle linear equations effectively.

Understanding the Basics of Linear Equations

Before diving into the specific equation, let's establish a solid foundation by understanding the fundamental concepts of linear equations. A linear equation is an algebraic equation in which the highest power of the variable is 1. These equations can be represented graphically as a straight line. The general form of a linear equation in one variable is $ax + b = c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. Solving a linear equation involves isolating the variable on one side of the equation to determine its value. This is achieved by performing the same operations on both sides of the equation, maintaining the equality. The operations typically involve addition, subtraction, multiplication, and division. Understanding these basic principles is essential for solving more complex linear equations and related problems.

Step-by-Step Solution of $5(3x+1) = 50$

Now, let's embark on the journey of solving the equation $5(3x+1) = 50$ step by step. We will meticulously explain each operation and the reasoning behind it, ensuring clarity and comprehension.

Step 1: Distribute the 5

The first step in solving this equation is to eliminate the parentheses by applying the distributive property. The distributive property states that $a(b + c) = ab + ac$ . In our equation, we need to distribute the 5 to both terms inside the parentheses: $5(3x + 1)$ .

Multiplying 5 by $3x$ gives us $15x$ , and multiplying 5 by 1 gives us 5. Therefore, the equation becomes:

$15x + 5 = 50$

This step simplifies the equation by removing the parentheses, making it easier to isolate the variable $x$ . Understanding and applying the distributive property is a crucial skill in algebra, as it allows us to handle expressions with parentheses effectively. This property is not only essential for solving linear equations but also for simplifying more complex algebraic expressions and equations.

Step 2: Subtract 5 from both sides

Our next goal is to isolate the term containing $x$ on one side of the equation. To do this, we need to eliminate the constant term, which is 5 in this case, from the left side. We can achieve this by subtracting 5 from both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must perform the same operation on the other side to maintain the equality.

Subtracting 5 from both sides, we get:

$15x + 5 - 5 = 50 - 5$

Simplifying, we have:

$15x = 45$

By subtracting 5 from both sides, we have successfully isolated the term with $x$ ( $15x$ ) on the left side of the equation. This step is a fundamental application of the properties of equality, which state that performing the same operation on both sides of an equation preserves the equality. This principle is crucial for solving any algebraic equation, as it allows us to manipulate the equation while maintaining its balance and ultimately finding the value of the variable.

Step 3: Divide both sides by 15

Now that we have $15x = 45$ , our final step is to isolate $x$ completely. Currently, $x$ is being multiplied by 15. To undo this multiplication, we need to perform the inverse operation, which is division. We will divide both sides of the equation by 15.

Dividing both sides by 15, we get:

$rac{15x}{15} = rac{45}{15}$

Simplifying, we find:

$x = 3$

Therefore, the solution to the equation $5(3x + 1) = 50$ is $x = 3$ . This final step demonstrates the power of inverse operations in solving equations. By dividing both sides by 15, we effectively "undid" the multiplication and isolated $x$ , revealing its value. This technique is a cornerstone of algebraic problem-solving and is applicable to a wide range of equations.

Verifying the Solution

After finding a solution, it's always a good practice to verify it. This ensures that our solution is correct and that we haven't made any errors in our calculations. To verify the solution, we substitute the value of $x$ back into the original equation and check if the equation holds true.

Substituting $x = 3$ into the original equation $5(3x + 1) = 50$ , we get:

$5(3(3) + 1) = 50$

Simplifying the expression inside the parentheses:

$5(9 + 1) = 50$

$5(10) = 50$

$50 = 50$

Since the equation holds true, our solution $x = 3$ is correct. Verification is a crucial step in the problem-solving process. It not only confirms the accuracy of our solution but also helps us identify any potential errors in our calculations or reasoning. By substituting the solution back into the original equation, we can gain confidence in our answer and ensure that we have solved the problem correctly.

Alternative Approaches to Solving the Equation

While we have solved the equation $5(3x + 1) = 50$ using the standard method of distribution and isolating the variable, there are alternative approaches that can be employed. Exploring these alternative methods can provide a deeper understanding of the equation and enhance problem-solving flexibility.

Method 1: Dividing by 5 First

Instead of distributing the 5 in the first step, we can divide both sides of the equation by 5. This is a valid operation because dividing both sides of an equation by the same non-zero number maintains the equality.

Dividing both sides of $5(3x + 1) = 50$ by 5, we get:

$rac{5(3x + 1)}{5} = rac{50}{5}$

Simplifying, we have:

$3x + 1 = 10$

Now, we can subtract 1 from both sides:

$3x + 1 - 1 = 10 - 1$

$3x = 9$

Finally, we divide both sides by 3:

$rac{3x}{3} = rac{9}{3}$

$x = 3$

This method yields the same solution, $x = 3$ , but it involves a different sequence of steps. By dividing by 5 first, we simplify the equation earlier in the process, potentially making the subsequent steps easier to manage. This alternative approach highlights the flexibility in solving linear equations and demonstrates that there can be multiple paths to the correct solution. Understanding these alternative methods can empower problem solvers to choose the most efficient approach based on the specific characteristics of the equation.

Comparing the Methods

Both methods are valid and lead to the correct solution. The choice of method often depends on personal preference or the specific structure of the equation. Some individuals may find it easier to distribute first, while others may prefer to simplify the equation by dividing first. The key is to understand the underlying principles and apply them correctly.

Real-World Applications of Linear Equations

Linear equations are not just abstract mathematical concepts; they have numerous real-world applications. Understanding how to solve linear equations is essential for various fields, including:

Finance: Calculating simple interest, loan payments, and investment returns.
Physics: Describing motion, calculating forces, and analyzing circuits.
Engineering: Designing structures, analyzing systems, and solving optimization problems.
Economics: Modeling supply and demand, analyzing market trends, and forecasting economic growth.
Everyday Life: Calculating expenses, budgeting finances, and making informed decisions.

For example, consider a scenario where you want to determine how many hours you need to work at a part-time job to earn enough money to buy a new phone. If you know your hourly wage and the cost of the phone, you can set up a linear equation to solve for the number of hours required. Similarly, linear equations can be used to calculate the distance traveled by a car at a constant speed, the amount of ingredients needed for a recipe, or the total cost of a purchase with sales tax. The ability to translate real-world problems into linear equations and solve them is a valuable skill that empowers individuals to make informed decisions and solve practical challenges.

Common Mistakes to Avoid

Solving linear equations involves following a systematic approach and applying the correct operations. However, it's easy to make mistakes if one is not careful. Here are some common mistakes to avoid:

Incorrect Distribution: Forgetting to distribute to all terms inside the parentheses.
Incorrect Sign: Making errors with positive and negative signs when adding or subtracting terms.
Incorrect Order of Operations: Not following the correct order of operations (PEMDAS/BODMAS).
Dividing by Zero: Dividing both sides of the equation by zero (which is undefined).
Not Verifying the Solution: Failing to substitute the solution back into the original equation to check for errors.

For instance, a common mistake is to distribute only to the first term inside the parentheses, resulting in an incorrect equation. Another frequent error is making sign mistakes when moving terms from one side of the equation to the other. To avoid these mistakes, it's crucial to practice careful attention to detail, double-check each step, and verify the solution. By being aware of these common pitfalls and taking steps to prevent them, one can significantly improve their accuracy and confidence in solving linear equations.

Practice Problems

To solidify your understanding and enhance your problem-solving skills, practice is essential. Here are some practice problems for you to try:

$2(x + 3) = 10$
$4x - 7 = 5$
$3(2x - 1) = 15$
$-2x + 8 = 2$
$rac{x}{2} + 5 = 9$

Solving these practice problems will reinforce the concepts and techniques discussed in this guide. By working through these examples, you will develop fluency in solving linear equations and gain confidence in your ability to tackle more complex problems. Practice is the key to mastering any mathematical skill, and linear equations are no exception. The more you practice, the more comfortable and proficient you will become in solving them.

Conclusion

Solving linear equations is a fundamental skill in mathematics with wide-ranging applications. This comprehensive guide has provided a step-by-step approach to solving the equation $5(3x + 1) = 50$ , along with explanations, alternative methods, and common mistakes to avoid. By understanding the underlying principles and practicing regularly, you can master the art of solving linear equations and apply this knowledge to various real-world scenarios. Whether you are a student, a professional, or simply someone interested in expanding your mathematical knowledge, this guide serves as a valuable resource for mastering linear equations. Remember, consistent practice and a solid understanding of the fundamentals are the keys to success in mathematics.