Solving For X: A Step-by-Step Guide To Algebraic Equations

by Alex Johnson 59 views

Are you ready to dive into the world of algebra and learn how to solve for x? In this comprehensive guide, we'll break down the equation -5(-5x + 2) + 3x + 5 = 23 step by step, making the process easy to understand, even if you're new to algebra. We'll explore the fundamental principles involved, ensuring you gain the confidence to tackle similar problems independently. Whether you're a student struggling with homework or someone looking to brush up on their math skills, this article provides all the necessary information to master solving algebraic equations.

Understanding the Basics: The Language of Algebra

Before we jump into solving the equation, let's establish a solid foundation in algebraic concepts. Algebra, at its core, is a language of symbols and rules that allows us to represent relationships between numbers and variables. A variable, usually represented by a letter like 'x', is a placeholder for an unknown value. Our goal in solving an equation is to isolate this variable and determine its value. Equations are mathematical statements that show two expressions are equal, linked by an equals sign (=). Our equation -5(-5x + 2) + 3x + 5 = 23 tells us that the expression on the left-hand side has the same value as the number on the right-hand side. The key to solving equations lies in manipulating them using specific rules while keeping the equation balanced. These rules are derived from the properties of arithmetic operations. These include the commutative, associative, and distributive properties, along with the properties of equality. These principles will act as our guiding rules throughout the equation-solving journey. Grasping these basics is vital for navigating the more complex steps that follow, and understanding the logic behind each step allows for a more intuitive approach to problem-solving. This makes the whole process more manageable and less intimidating.

Order of Operations: PEMDAS/BODMAS

Remembering the order of operations is also important when solving algebraic equations. We follow the PEMDAS rule, which is a handy acronym that stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In some regions, BODMAS is used, which is similar but replaces Parentheses with Brackets. This order provides a structured approach, helping us tackle the equation in the right sequence. The priority is to address anything inside parentheses or brackets first, then handle exponents, followed by multiplication and division (working left to right), and finally, addition and subtraction (also working left to right). By adhering to these steps, we ensure consistency and arrive at the accurate solution. Understanding these rules is essential as they dictate the order in which we should tackle all parts of the equation, making the process logical and systematic. We will come back to this concept while solving our equation below, applying each step in a methodical way.

Step-by-Step Solution: Unraveling the Equation

Now, let's take on the equation -5(-5x + 2) + 3x + 5 = 23 and unravel it step by step. We'll apply the principles of algebra to systematically isolate 'x' and determine its value. Let's start with simplifying and removing parentheses using the distributive property. Then we will combine like terms on the same side of the equation. Finally, we'll isolate the variable 'x' to find the solution.

Step 1: Distribute and Simplify

The first step involves applying the distributive property to remove the parentheses. This means multiplying the number outside the parentheses by each term inside the parentheses. So, for -5(-5x + 2), we multiply -5 by -5x and then by +2. This step is crucial because it simplifies the expression by getting rid of the parentheses, and makes it easier to combine like terms. This gives us:

-5 * -5x = 25x

-5 * 2 = -10

Therefore, the equation becomes:

25x - 10 + 3x + 5 = 23

Step 2: Combine Like Terms

Next, we'll combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, 25x and 3x are like terms and -10 and 5 are like terms. Combining these terms simplifies the equation, making it easier to solve. Grouping these terms together streamlines the equation and prepares us for isolating the variable. Adding like terms ensures we're dealing with fewer terms and simplifies the equation.

Combining 25x and 3x gives us 28x and combining -10 and 5 gives us -5. So, the equation now becomes:

28x - 5 = 23

Step 3: Isolate the Variable

Our next goal is to isolate the variable 'x'. To do this, we need to move the constant term -5 to the other side of the equation. We do this by performing the opposite operation. In this case, we add 5 to both sides of the equation. It's really important to perform the same operation on both sides to keep the equation balanced.

28x - 5 + 5 = 23 + 5

This simplifies to:

28x = 28

Step 4: Solve for x

Finally, we solve for 'x' by dividing both sides of the equation by the coefficient of 'x', which is 28. This isolates 'x' and reveals its value. Dividing both sides by the same number keeps the equation balanced, allowing us to find the accurate solution.

28x / 28 = 28 / 28

This simplifies to:

x = 1

Verification: Checking Our Work

To ensure our solution is correct, we can substitute x = 1 back into the original equation and check if both sides are equal. This step is a vital habit because it confirms that our solution satisfies the equation and that all steps were executed correctly. Substitute 'x' with '1' and perform the operations in the correct order. This helps to catch any errors and ensures the accuracy of our final answer. Substitute x = 1 into the original equation:

-5(-5 * 1 + 2) + 3 * 1 + 5 = 23

-5(-5 + 2) + 3 + 5 = 23

-5(-3) + 3 + 5 = 23

15 + 3 + 5 = 23

23 = 23

Since both sides are equal, our solution x = 1 is correct.

Common Mistakes and How to Avoid Them

Understanding common pitfalls can help prevent errors and enhance problem-solving skills. The most typical mistakes include errors in the order of operations, incorrect application of the distributive property, and making arithmetic errors when combining like terms. Being mindful of these potential issues is a great way to improve your accuracy.

  • Order of Operations Errors: Always follow the PEMDAS/BODMAS rule. Incorrect order leads to the wrong answer. Double-check each step. When evaluating each section of the equation, use the correct order. This is a very common error and can derail even the simplest problems.
  • Incorrect Distribution: Ensure you multiply the term outside the parentheses with each term inside. Often, students forget to multiply by all the terms, which can throw off the calculation. Carefully double-check and perform each step to avoid errors.
  • Arithmetic Errors: Carelessly adding, subtracting, multiplying, or dividing can result in incorrect answers. Always re-evaluate each calculation. Review your calculations and use a calculator if required, but always double-check the results by hand.
  • Forgetting to Apply Operations to Both Sides: It is important to remember that whatever operations you perform on one side must be performed on the other to keep the equation balanced. This is a crucial rule.

Conclusion: Mastering the Art of Solving for x

Solving for 'x' in algebraic equations is a fundamental skill that opens the door to more complex mathematical concepts. By breaking down the equation -5(-5x + 2) + 3x + 5 = 23 step by step, we've demonstrated how to approach such problems systematically. Remember to practice regularly, to apply the principles we discussed, and to stay patient. With dedication and attention to detail, you can master solving for x. The process becomes easier with each solved equation. Keep practicing, reviewing the steps, and, most importantly, don’t be afraid to make mistakes. Mistakes provide opportunities for learning and further improvement.

This detailed guide equips you with the necessary knowledge and techniques to solve algebraic equations. The methodical process ensures a solid grasp of concepts, leading to increased confidence in tackling complex problems. By understanding the underlying principles and practicing regularly, you can successfully solve for x and achieve success in mathematics.

For additional practice and deeper understanding, you might find the resources on Khan Academy helpful.