Solving Equations: Step-by-Step Guide For 3.5 + 1.2(6.3 - 7x)

Let's break down how to solve the equation 3.5 + 1.2(6.3 - 7x) = 9.38. If you've ever felt a little lost staring at an equation, don't worry! We're going to take it one step at a time, making it super clear and easy to follow. Solving equations might seem intimidating at first, but with a methodical approach, anyone can master it. This guide will walk you through each step involved in solving this particular equation, ensuring you understand the logic behind every move. So, grab your pencil and paper, and let's get started!

Understanding the Order of Operations

Before we dive into the specific steps for this equation, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order tells us which operations to perform first. It's the golden rule of equation solving! Neglecting this order can lead to incorrect solutions, which is why mastering PEMDAS is the first key to unlocking success in algebra. Remember, solving equations is like following a recipe – you need to add the ingredients in the right order to get the desired result.

Why PEMDAS Matters

Imagine if we didn't have a standard order of operations. Different people might solve the same equation in different ways and arrive at various answers. PEMDAS ensures everyone follows the same rules, leading to consistent and accurate results. This standardization is crucial in mathematics and other fields that rely on mathematical calculations. For instance, in computer programming, the order of operations dictates how a program interprets mathematical expressions. Misinterpreting this order could lead to significant errors in software functionality, highlighting the practical importance of PEMDAS beyond the classroom.

Possible Steps to Solve the Equation

Now, let's look at the equation 3.5 + 1.2(6.3 - 7x) = 9.38 and identify the correct steps to solve for 'x'. We'll go through each option, explaining why it's either a correct step or not. The beauty of algebra lies in its structured approach, where each step logically builds upon the previous one. To effectively solve equations, it’s not just about knowing the rules, but also understanding why these rules work. This deeper understanding will empower you to tackle more complex problems with confidence. So, let's put on our detective hats and solve this equation together!

Step 1: Distribute 1.2 to 6.3 and -7x

This is the correct first step according to PEMDAS. We need to eliminate the parentheses before we can combine any terms. Distribution involves multiplying the term outside the parentheses (1.2) by each term inside the parentheses (6.3 and -7x). This process is fundamental to simplifying algebraic expressions and equations. It's like unpacking a box – you need to take out each item individually before you can organize them. Distribution not only helps to simplify equations but also provides insights into the relationships between different parts of an expression. Mastering this step is crucial for handling more complex algebraic manipulations.

• 1.2 * 6.3 = 7.56
• 1.2 * -7x = -8.4x

So, after distributing, the equation becomes: 3.5 + 7.56 - 8.4x = 9.38. Distribution is a powerful tool that transforms equations into a more manageable form. It allows us to separate terms that are initially grouped together, making it easier to isolate the variable and ultimately solve the equation. By understanding the mechanics and purpose of distribution, you can navigate complex algebraic landscapes with greater ease and precision.

Step 2: Combine 3.5 and 7.56

Now that we've distributed, we can combine the constant terms on the left side of the equation. This simplifies the equation further and brings us closer to isolating the variable 'x'. Combining like terms is a core concept in algebra, akin to sorting similar items into categories. It streamlines the equation, making it easier to see the relationships between the terms. This step also reinforces the importance of maintaining balance in the equation – what you do on one side, you must also do on the other. By mastering the art of combining like terms, you develop a critical skill for solving a wide range of algebraic problems.

• 3. 5 + 7.56 = 11.06

The equation now looks like this: 11.06 - 8.4x = 9.38. Combining like terms is a fundamental algebraic technique that simplifies equations and prepares them for further steps in the solution process. It's a skill that is frequently used in various branches of mathematics, making it an indispensable part of your algebraic toolkit. Proficiency in combining like terms can significantly enhance your ability to tackle more complex mathematical problems with confidence and efficiency.

Step 3: Subtract 11.06 from Both Sides

To isolate the term with 'x', we need to subtract 11.06 from both sides of the equation. This maintains the balance of the equation while moving us closer to solving for 'x'. The principle of maintaining balance is paramount in equation solving, akin to a seesaw that must remain level. Any operation performed on one side must be mirrored on the other side to preserve the equation's integrity. This step showcases the fundamental concept of inverse operations – using subtraction to undo addition – a cornerstone of algebraic manipulation. By consistently applying this principle, you ensure the accuracy of your solutions and deepen your understanding of equation solving.

• 11. 06 - 8.4x - 11.06 = 9.38 - 11.06
• -8.4x = -1.68

Subtracting from both sides is a key step in isolating the variable. This move is based on the fundamental principle of equality, which states that if you perform the same operation on both sides of an equation, the equation remains balanced. Mastering this technique is crucial for solving a wide array of equations and is a foundational skill in algebra.

Step 4: Divide Both Sides by -8.4

Finally, to solve for 'x', we divide both sides of the equation by -8.4. This isolates 'x' and gives us the solution. Division, as the inverse operation of multiplication, plays a crucial role in isolating variables in algebraic equations. This step highlights the power of inverse operations in reversing mathematical processes and solving for unknowns. It also underscores the importance of paying attention to signs – dividing a negative number by another negative number yields a positive result. This final step not only provides the solution but also reinforces the concept of inverse operations and their application in solving equations. By understanding and applying this step, you complete the process of isolating the variable and finding its value.

• -8. 4x / -8.4 = -1.68 / -8.4
• x = 0.2

Dividing both sides by the coefficient of 'x' is the final step in isolating the variable. This operation reveals the value of 'x' that satisfies the original equation. It's the culmination of all the previous steps, demonstrating the power of systematic algebraic manipulation. The solution, x = 0.2, represents the point where the equation holds true, showcasing the precision and elegance of mathematical problem-solving.

Incorrect Options Explained

Let's also briefly discuss why the other options listed in the original question are incorrect:

• A. Add 3.5 and 1.2: This is incorrect because we need to address the parentheses first due to the order of operations.
• C. Combine 6.3 and -7x: These are not like terms and cannot be combined directly.
• D. Combine 3.5 and 7.56: While this step is correct, it comes after distributing 1.2.
• E. Subtract 11.06 from both sides: This step is also correct but needs to happen after combining the constant terms.

Conclusion

Solving equations involves a systematic approach, and understanding the order of operations (PEMDAS) is key. By following the correct steps – distributing, combining like terms, and using inverse operations – you can successfully solve for the unknown variable. Remember, practice makes perfect! The more you solve equations, the more comfortable and confident you'll become. Math, at its core, is about problem-solving. By breaking down complex problems into manageable steps, we can unravel the mysteries hidden within equations. This process not only enhances our mathematical skills but also cultivates critical thinking and logical reasoning, skills that extend far beyond the realm of mathematics. So, keep practicing, keep exploring, and keep embracing the power of problem-solving!

For more in-depth information and practice problems, check out resources like Khan Academy's Algebra Section.