Verifying Solutions Is 11 A Solution Of 7(x-1)=70

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Introduction

In mathematics, determining whether a given value is a solution to an equation is a fundamental skill. It involves substituting the value into the equation and checking if the resulting statement is true. In this article, we will delve into the process of verifying if 11 is a solution to the equation 7(x-1)=70. We will explore the steps involved, discuss the underlying mathematical principles, and provide a comprehensive understanding of how to solve such problems. This exploration is not just an exercise in algebra; it's a cornerstone of mathematical understanding, applicable in various fields from engineering to economics. Understanding how to verify solutions helps build a strong foundation for more advanced mathematical concepts and real-world problem-solving.

Understanding the Equation

Before we jump into the verification process, let's first understand the equation itself. The equation 7(x-1)=70 is a linear equation in one variable, 'x'. Linear equations are algebraic expressions where the highest power of the variable is 1. They represent a straight line when graphed on a coordinate plane, making them a fundamental concept in algebra. The equation consists of two sides: the left-hand side (LHS), which is 7(x-1), and the right-hand side (RHS), which is 70. The goal is to find the value(s) of 'x' that make the LHS equal to the RHS. This value(s) is called the solution(s) of the equation. Understanding the structure of the equation is crucial because it dictates how we approach solving it. The presence of parentheses on the LHS indicates that the distributive property will likely be involved, while the constant term on the RHS provides a target value for our algebraic manipulations. The equation 7(x-1)=70 is a simple yet powerful example of how algebraic equations can model real-world scenarios, from calculating costs to determining distances.

Step-by-Step Verification Process

To determine if 11 is a solution to the equation 7(x-1)=70, we follow a straightforward step-by-step process:

  1. Substitution: The first step is to substitute the value 11 for 'x' in the equation. This means replacing every instance of 'x' in the equation with the number 11. So, the equation 7(x-1)=70 becomes 7(11-1)=70. This substitution transforms the equation from an algebraic statement into an arithmetic one, which can be directly evaluated.
  2. Simplification: Next, we simplify the equation using the order of operations (PEMDAS/BODMAS). This involves performing the operations within the parentheses first. In this case, we subtract 1 from 11, which gives us 10. So, the equation becomes 7(10)=70. Then, we multiply 7 by 10, which results in 70. Therefore, the left-hand side of the equation simplifies to 70.
  3. Comparison: Finally, we compare the simplified left-hand side (LHS) with the right-hand side (RHS) of the equation. If the LHS is equal to the RHS, then the value we substituted (in this case, 11) is a solution to the equation. If the LHS is not equal to the RHS, then the value is not a solution. In our case, the LHS simplifies to 70, which is equal to the RHS, which is also 70. Thus, we can conclude that 11 is indeed a solution to the equation 7(x-1)=70.

Detailed Solution

Let's walk through the solution in detail:

  • Original Equation: 7(x-1) = 70
  • Substitute x = 11: 7(11-1) = 70
  • Simplify inside the parentheses: 7(10) = 70
  • Multiply: 70 = 70

Since the left-hand side (70) is equal to the right-hand side (70), the equation holds true when x = 11. This confirms that 11 is a solution to the equation 7(x-1) = 70. This detailed step-by-step approach ensures clarity and precision in our mathematical reasoning. By breaking down the problem into smaller, manageable steps, we can confidently arrive at the correct conclusion. This method is applicable to a wide range of algebraic problems and highlights the importance of methodical problem-solving in mathematics.

Why Does This Method Work?

The method of substituting a value into an equation and checking for equality works because it's based on the fundamental definition of a solution. A solution to an equation is a value that, when substituted for the variable, makes the equation true. In other words, it's a value that balances the equation, making the left-hand side equal to the right-hand side. When we substitute 11 for 'x' in the equation 7(x-1)=70, we are essentially testing whether 11 is such a balancing value. By simplifying both sides of the equation after the substitution, we are determining whether the two sides are indeed equal when x=11. If they are equal, then 11 satisfies the equation's condition and is therefore a solution. This method is a direct application of the definition of a solution and is a cornerstone of algebraic problem-solving. It allows us to verify potential solutions and ensure that they satisfy the given equation. The principle of maintaining balance in an equation is a crucial concept in algebra and is used in various techniques, including solving equations and inequalities.

Alternative Methods for Solving the Equation

While substituting and verifying is a direct method, it's also beneficial to know how to solve the equation from scratch. This not only confirms our answer but also deepens our understanding of algebraic manipulation. There are a couple of ways to solve the equation 7(x-1)=70:

  1. Distributive Property: We can start by applying the distributive property to the left-hand side of the equation. This means multiplying the 7 by both terms inside the parentheses: 7 * x - 7 * 1 = 7x - 7. So, the equation becomes 7x - 7 = 70. Next, we isolate the term with 'x' by adding 7 to both sides of the equation: 7x - 7 + 7 = 70 + 7, which simplifies to 7x = 77. Finally, we solve for 'x' by dividing both sides of the equation by 7: 7x / 7 = 77 / 7, which gives us x = 11.
  2. Dividing First: Another approach is to divide both sides of the original equation by 7 before applying the distributive property. This simplifies the equation and reduces the numbers we are working with. Dividing both sides of 7(x-1)=70 by 7 gives us (x-1) = 10. Then, we isolate 'x' by adding 1 to both sides of the equation: x - 1 + 1 = 10 + 1, which simplifies to x = 11. Both methods lead to the same solution, x = 11, confirming our earlier verification. Understanding different methods for solving equations is crucial because it allows us to choose the most efficient approach for a given problem and reinforces our understanding of algebraic principles. These alternative methods not only provide a different perspective on the problem but also enhance our problem-solving skills in mathematics.

Common Mistakes and How to Avoid Them

When verifying or solving equations, several common mistakes can occur. Recognizing these errors and understanding how to avoid them is crucial for accuracy. One frequent mistake is incorrect substitution. For example, students might substitute 11 for 'x' only in one part of the equation or misinterpret the equation's structure. To avoid this, always double-check that you've replaced every instance of the variable with the given value. Another common mistake is errors in simplification, particularly with the order of operations. For instance, students might add 7 to 70 before distributing the 7, which is incorrect. Remember to follow the order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Another mistake is incorrect application of the distributive property. Make sure to multiply the term outside the parentheses by every term inside the parentheses. Additionally, errors can occur during the process of isolating the variable, such as adding or subtracting terms incorrectly on both sides of the equation. To avoid these mistakes, always perform the same operation on both sides of the equation to maintain balance. Finally, a simple but important step is to double-check your work. After solving an equation, substitute your solution back into the original equation to verify that it holds true. By being aware of these common mistakes and adopting careful problem-solving strategies, you can significantly improve your accuracy in algebra.

Real-World Applications

Understanding how to solve and verify equations is not just an academic exercise; it has numerous real-world applications. Equations are used to model and solve problems in various fields, from finance to engineering. For example, in finance, equations can be used to calculate loan payments, investment returns, or budget constraints. Verifying solutions is crucial in this context to ensure the accuracy of financial calculations and decisions. In engineering, equations are used to design structures, circuits, and systems. Engineers need to verify that their solutions meet specific requirements and constraints. For instance, an equation might be used to calculate the load-bearing capacity of a bridge, and the solution must be verified to ensure the bridge's safety. In physics, equations are used to describe motion, energy, and forces. Verifying solutions is essential for predicting the behavior of physical systems and designing experiments. Even in everyday life, we use equations and problem-solving skills to make decisions, such as calculating the cost of a purchase or determining the time needed to travel a certain distance. The ability to verify solutions ensures that our decisions are based on accurate information. Therefore, mastering the skills of solving and verifying equations is not only important for academic success but also for navigating the complexities of the real world. By understanding how mathematical principles apply to everyday situations, we can make more informed decisions and solve problems more effectively.

Conclusion

In conclusion, we have thoroughly demonstrated that 11 is indeed a solution to the equation 7(x-1)=70. We achieved this by substituting 11 for 'x' in the equation and verifying that the resulting statement is true. We also explored alternative methods for solving the equation, which further confirmed our solution. This process underscores the importance of understanding the fundamental principles of algebra, such as the order of operations, the distributive property, and the concept of balancing equations. By mastering these skills, we can confidently solve a wide range of algebraic problems and apply them to real-world situations. Furthermore, we discussed common mistakes to avoid, emphasizing the need for careful attention to detail and methodical problem-solving. The ability to verify solutions is a crucial skill that ensures accuracy and builds confidence in our mathematical abilities. Ultimately, this exploration highlights the power and versatility of algebra as a tool for problem-solving and decision-making in various fields. The ability to manipulate equations, verify solutions, and apply mathematical principles to real-world scenarios is a valuable asset that empowers us to tackle complex challenges and make informed decisions.