Solving For X Find The Value In 5x + 13° = 88°

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Introduction

In the realm of mathematics, solving equations is a fundamental skill. This article delves into a specific algebraic problem: finding the value of 'x' in the equation 5x + 13° = 88°. This seemingly simple equation provides a great opportunity to illustrate the basic principles of algebra, including isolating variables and performing arithmetic operations to find solutions. We will walk through the step-by-step solution, discuss the concepts involved, and highlight the importance of accuracy in mathematical calculations. Whether you're a student learning algebra for the first time or someone looking to refresh your math skills, this guide offers a clear and concise approach to solving linear equations.

Problem Statement: 5x + 13° = 88°

The problem at hand is to determine the value of the variable 'x' that satisfies the equation 5x + 13° = 88°. This is a linear equation, characterized by having the variable 'x' raised to the power of 1. Solving for 'x' involves isolating the variable on one side of the equation by performing inverse operations. We will meticulously go through each step to ensure a clear understanding of the process. The equation itself is a statement of equality between two expressions, and our goal is to find the specific value of 'x' that makes this statement true. This process is a cornerstone of algebra and is used extensively in various fields of science, engineering, and economics. The ability to confidently solve such equations is a crucial skill for any aspiring mathematician or problem-solver.

Step-by-Step Solution

Step 1: Isolate the Term with 'x'

Our initial equation is 5x + 13° = 88°. The first step in solving for 'x' is to isolate the term that contains 'x', which is '5x'. To do this, we need to eliminate the constant term, '+13°', from the left side of the equation. We accomplish this by performing the inverse operation of addition, which is subtraction. Specifically, we subtract 13° from both sides of the equation. This maintains the balance of the equation, ensuring that the left side remains equal to the right side. The equation now becomes: 5x + 13° - 13° = 88° - 13°. Simplifying this, we get 5x = 75°. This step is crucial as it brings us closer to isolating 'x' and finding its value. By subtracting 13° from both sides, we have effectively moved the constant term to the right side of the equation, leaving the term with 'x' alone on the left side.

Step 2: Solve for 'x'

Having isolated the term '5x', our equation now reads 5x = 75°. To finally solve for 'x', we need to undo the multiplication by 5. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 5. This ensures that the equation remains balanced and that the value of 'x' remains unchanged. Dividing both sides by 5 gives us: (5x) / 5 = 75° / 5. Simplifying this, we find x = 15°. This is the solution to the equation, meaning that when x is equal to 15°, the original equation 5x + 13° = 88° holds true. This step is the culmination of the algebraic process, where we have successfully isolated 'x' and determined its numerical value. The result, x = 15°, is the answer to the problem, and it represents the value that satisfies the given equation.

Final Answer

Therefore, the value of x in the equation 5x + 13° = 88° is 15°. This solution was obtained by systematically isolating the variable 'x' through the application of inverse operations. First, we subtracted 13° from both sides of the equation to isolate the term '5x'. Then, we divided both sides by 5 to solve for 'x'. This step-by-step approach demonstrates the fundamental principles of algebra and the importance of maintaining balance in equations. The final answer, x = 15°, is the unique value that satisfies the given equation, making it a solution. Understanding and applying these algebraic techniques is essential for solving a wide range of mathematical problems and for further studies in mathematics and related fields.

Why is this Important?

Understanding how to solve equations like 5x + 13° = 88° is a foundational skill in mathematics. It's not just about finding the value of 'x'; it's about grasping the underlying principles of algebra and problem-solving. These skills are crucial for various reasons:

  • Foundation for Advanced Math: Solving linear equations is a building block for more complex algebraic concepts like quadratic equations, systems of equations, and calculus. Without a solid understanding of these basics, progressing in mathematics becomes significantly more challenging.
  • Real-World Applications: Algebra isn't confined to textbooks; it's used extensively in real-world scenarios. From calculating finances and budgeting to designing structures and predicting outcomes in science, algebraic thinking is essential.
  • Critical Thinking and Problem-Solving: Solving equations requires logical reasoning and problem-solving skills. By breaking down a problem into smaller steps and applying appropriate operations, you develop a systematic approach that can be applied to various challenges in life.
  • Analytical Skills: The process of isolating variables and manipulating equations hones your analytical skills. You learn to identify patterns, relationships, and dependencies, which are valuable in many fields.
  • Career Opportunities: Many careers, including engineering, computer science, finance, and data analysis, rely heavily on algebraic skills. Being proficient in solving equations can open doors to a wide range of opportunities.

Common Mistakes to Avoid

When solving algebraic equations, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Incorrect Order of Operations: Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Failing to follow this order can lead to incorrect results.
  • Not Performing the Same Operation on Both Sides: The golden rule of algebra is to maintain balance in the equation. Any operation you perform on one side must also be performed on the other side. For example, if you subtract 13° from the left side, you must also subtract 13° from the right side.
  • Sign Errors: Pay close attention to the signs (+ and -) of the terms. A simple sign error can completely change the outcome of the equation.
  • Dividing by Zero: Division by zero is undefined and leads to mathematical errors. Be careful not to divide by zero at any point in your solution.
  • Incorrectly Combining Like Terms: Only combine terms that have the same variable and exponent. For example, you can combine 5x and 3x, but you cannot combine 5x and 3x². Make sure you are combining the coefficients correctly while considering the signs.

Conclusion

In summary, finding the value of 'x' in the equation 5x + 13° = 88° is a straightforward algebraic problem that highlights the fundamental principles of equation solving. By isolating the variable 'x' through the application of inverse operations—subtracting 13° from both sides and then dividing by 5—we arrive at the solution x = 15°. This exercise underscores the importance of maintaining balance in equations and the meticulous application of arithmetic operations. Mastering these basic algebraic skills is crucial for tackling more complex mathematical challenges and for applying mathematical concepts in various real-world contexts. Whether you are a student learning algebra or someone seeking to refresh your mathematical abilities, understanding and practicing these techniques will undoubtedly enhance your problem-solving skills and mathematical proficiency.