Solving Equations Step-by-Step 3.5 + 1.2(6.3 - 7x) = 9.38

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When faced with an equation like 3.5 + 1.2(6.3 - 7x) = 9.38, it's crucial to approach it systematically to arrive at the correct solution. This equation involves several arithmetic operations, including addition, subtraction, multiplication, and the distributive property. To effectively solve for the unknown variable, 'x', we need to follow the correct order of operations and apply algebraic principles. Let's break down the possible steps involved in solving this equation, highlighting the key concepts and techniques that will lead us to the answer.

Understanding the Order of Operations

Before diving into the specific steps for this equation, it's essential to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform operations within an equation or expression. In our case, we have parentheses, multiplication, addition, and subtraction, so we'll need to adhere to PEMDAS to ensure we solve the equation correctly.

Step-by-Step Solution

Let's explore the possible steps involved in solving the equation 3.5 + 1.2(6.3 - 7x) = 9.38, examining each option and explaining why it's either a valid or invalid step.

Option A Add 3. 5 and 1. 2

This option suggests adding 3.5 and 1.2 directly. However, this is incorrect according to the order of operations. PEMDAS tells us that we must address the operations within the parentheses and any multiplication before we can perform addition. The 1.2 is multiplied by the entire expression within the parentheses (6.3 - 7x), so we cannot simply add it to 3.5 at this stage. Adding 3.5 and 1.2 prematurely would violate the order of operations and lead to an incorrect solution. It’s important to isolate the term being multiplied by the parentheses before performing any addition or subtraction outside of the parentheses.

Option B Distribute 1. 2 to 6. 3 and -7x

This option is correct. The distributive property states that a(b + c) = ab + ac. In our equation, 1.2 is multiplied by the expression (6.3 - 7x), so we need to distribute the 1.2 to both terms inside the parentheses. This means we multiply 1.2 by 6.3 and 1.2 by -7x. This step is crucial for simplifying the equation and isolating the variable 'x'. By distributing, we eliminate the parentheses and create individual terms that can be combined and manipulated further.

The distributive property is a fundamental concept in algebra, allowing us to expand expressions and make them easier to solve. In this case, distributing 1.2 allows us to remove the parentheses and work with individual terms, which is a necessary step in solving for 'x'. The result of this distribution will be two separate terms: one involving 'x' and one constant term. This sets the stage for further simplification and isolation of the variable.

Option C Combine 6. 3 and -7x

This option is incorrect. We cannot combine 6.3 and -7x directly because they are not like terms. Like terms have the same variable raised to the same power. 6.3 is a constant term, while -7x is a term with the variable 'x'. We can only combine terms that have the same variable and exponent. Attempting to combine these unlike terms would be a mathematical error and would not help us in solving the equation. The expression (6.3 - 7x) represents a difference between a constant and a variable term, and these cannot be combined into a single term.

Option D Combine 3. 5 and the result of distributing 1. 2

This option is correct. After distributing 1.2 to both terms inside the parentheses, we will have a new equation with several terms. At this point, we can combine the constant terms, including the 3.5. This involves adding 3.5 to the constant term that results from the distribution (1.2 * 6.3). Combining like terms is a key step in simplifying equations and bringing us closer to isolating the variable. By combining the constant terms, we reduce the number of terms in the equation and make it easier to manipulate algebraically.

Combining like terms is a fundamental algebraic technique that simplifies equations by grouping together terms with the same variable and exponent. In this case, we combine the constant terms to streamline the equation and prepare it for further steps in solving for 'x'. This process helps to consolidate the numerical components of the equation, making it easier to isolate the variable.

Detailed Breakdown of the Correct Steps

To further illustrate the correct steps, let's perform the actual calculations:

  1. Distribute 1.2 to 6.3 and -7x:

  2. 2 * 6.3 = 7.56

  3. 2 * -7x = -8.4x

The equation now becomes: 3.5 + 7.56 - 8.4x = 9.38

  1. Combine 3.5 and 7.56:

  2. 5 + 7.56 = 11.06

The equation now becomes: 11.06 - 8.4x = 9.38

From here, we would continue solving for 'x' by isolating the variable term and then dividing to find the value of 'x'. However, the initial steps of distributing and combining like terms are crucial for setting up the equation for the remaining steps.

Key Takeaways

Solving equations requires a systematic approach, and understanding the order of operations is paramount. In the equation 3.5 + 1.2(6.3 - 7x) = 9.38, the correct first steps involve:

  • Distributing 1.2 to both terms inside the parentheses (6.3 and -7x).
  • Combining the constant terms after the distribution (3.5 and the result of 1.2 * 6.3).

By following these steps, we can simplify the equation and pave the way for solving for the unknown variable 'x'. Remember to always adhere to PEMDAS and combine only like terms to ensure accuracy in your solutions.

Conclusion

In conclusion, when solving the equation 3.5 + 1.2(6.3 - 7x) = 9.38, the critical initial steps are to distribute the 1.2 across the terms within the parentheses and then combine the resulting constant term with the 3.5. These actions correctly apply the order of operations and simplify the equation, making it possible to isolate 'x' and determine its value. Avoiding the premature addition of 3.5 and 1.2, or the incorrect combination of 6.3 and -7x, ensures adherence to mathematical principles and leads to an accurate solution. Understanding and applying these steps are fundamental to solving algebraic equations effectively.