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

In the realm of mathematics, solving equations is a fundamental skill. It's like deciphering a secret code, where the unknown variable is the hidden message. This article will serve as your guide through the process of solving the equation 3.5 + 1.2(6.3 - 7x) = 9.38, providing a clear, step-by-step approach that will empower you to tackle similar problems with confidence. We will delve into the essential steps involved, highlighting the underlying principles that make equation solving such a powerful tool.

Understanding the Equation

Before we dive into the solution, let's take a moment to understand the equation we're dealing with: 3. 5 + 1.2(6.3 - 7x) = 9.38. This equation involves a variable, x, which represents an unknown value that we need to determine. The equation states that the expression on the left-hand side is equal to the value on the right-hand side, which is 9.38. To solve for x, our goal is to isolate it on one side of the equation. This means we need to manipulate the equation using algebraic operations until we have x by itself.

The equation contains several elements: constants (numbers like 3.5, 1.2, 6.3, and 9.38), a variable (x), and mathematical operations (addition, subtraction, and multiplication). The presence of parentheses indicates that we need to apply the distributive property, which we'll discuss in detail later. By carefully analyzing these components, we can devise a strategic approach to solving for x. Remember, the key to solving equations lies in maintaining balance. Whatever operation we perform on one side of the equation, we must also perform on the other side to ensure the equality remains valid. This principle of balance is the foundation of equation solving and will guide us through each step of the process.

Step 1 Distribute 1.2 to 6.3 and -7x

The initial step in solving the equation 3. 5 + 1.2(6.3 - 7x) = 9.38 involves applying the distributive property. This property is crucial when dealing with expressions that contain parentheses, as it allows us to simplify the equation by eliminating the parentheses. 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 1.2 to both 6.3 and -7x.

Let's break down the distribution process. First, we multiply 1.2 by 6.3, which yields 7.56. Next, we multiply 1.2 by -7x, resulting in -8.4x. Applying the distributive property transforms the equation from 3. 5 + 1.2(6.3 - 7x) = 9.38 to 3.5 + 7.56 - 8.4x = 9.38. This step is crucial because it removes the parentheses, making the equation easier to manipulate. By distributing the 1.2, we've effectively expanded the expression, revealing the individual terms that we can now combine and isolate. Remember, the order of operations (PEMDAS/BODMAS) dictates that we perform multiplication before addition or subtraction, making distribution a necessary first step in this case. The distributive property is a cornerstone of algebra, and mastering its application is essential for solving a wide range of equations.

Step 2 Combine Like Terms

After applying the distributive property, our equation now reads 3. 5 + 7.56 - 8.4x = 9.38. The next step is to combine like terms. Like terms are those that have the same variable raised to the same power. In this equation, 3.5 and 7.56 are like terms because they are both constants (numbers without any variables). Combining like terms simplifies the equation by reducing the number of terms, making it easier to isolate the variable.

To combine 3.5 and 7.56, we simply add them together: 3.5 + 7.56 = 11.06. This simplifies our equation to 11. 06 - 8.4x = 9.38. Notice that we cannot combine 11.06 with -8.4x because -8.4x contains the variable x, while 11.06 is a constant. Combining like terms is a fundamental algebraic technique that streamlines the equation-solving process. By grouping similar terms, we reduce complexity and make the equation more manageable. This step sets the stage for isolating the variable, which is our ultimate goal. It's important to remember that only like terms can be combined, and this principle applies to equations of all complexities.

Step 3 Isolate the Variable Term

With our equation simplified to 11. 06 - 8.4x = 9.38, the next crucial step is to isolate the variable term, which in this case is -8.4x. To isolate a term, we need to undo any operations that are being performed on it. In this instance, -8.4x is being added to 11.06 (remember that subtraction is the same as adding a negative number). To undo this addition, we need to subtract 11.06 from both sides of the equation.

Subtracting 11.06 from both sides maintains the balance of the equation, a fundamental principle of algebra. This gives us 11. 06 - 8.4x - 11.06 = 9.38 - 11.06. On the left side, 11.06 and -11.06 cancel each other out, leaving us with -8.4x. On the right side, 9.38 - 11.06 equals -1.68. Our equation is now simplified to -8.4x = -1.68. Isolating the variable term is a key step in solving equations because it brings us closer to determining the value of the variable. By strategically applying inverse operations, we can systematically peel away the layers surrounding the variable until it stands alone.

Step 4 Solve for the Variable

Having isolated the variable term, our equation is now -8. 4x = -1.68. The final step in solving for x is to undo the multiplication. The variable x is being multiplied by -8.4. To isolate x, we need to divide both sides of the equation by -8.4. Dividing both sides by the same number maintains the balance of the equation, ensuring that the equality remains valid.

Dividing both sides by -8.4 gives us (-8.4x) / -8.4 = (-1.68) / -8.4. On the left side, -8.4 divided by -8.4 equals 1, so we are left with x. On the right side, -1.68 divided by -8.4 equals 0.2. Therefore, the solution to the equation is x = 0.2. This final step is the culmination of all our previous efforts. By carefully applying algebraic principles, we have successfully isolated the variable and determined its value. The solution, x = 0.2, is the answer to our original question. It's the value that, when substituted back into the original equation, makes the equation true.

Selecting the Correct Options

Based on the steps we've taken, let's revisit the options provided and identify the correct ones:

A. Add 3.5 and 1.2. - Incorrect. We don't add 3.5 and 1.2 directly because 1.2 is multiplied by the expression in parentheses.

B. Distribute 1.2 to 6.3 and -7x. - Correct. This is the first step we took, applying the distributive property.

C. Combine 6.3 and -7x. - Incorrect. We cannot combine these terms because they are not like terms. 6.3 is a constant, while -7x contains the variable x.

D. Combine 3.5 and the result of distributing 1.2 to 6.3. - Correct. After distributing, we combine the constant terms.

Therefore, the correct options are B and D.

Conclusion

Solving equations is a fundamental skill in mathematics, and the equation 3. 5 + 1.2(6.3 - 7x) = 9.38 provides a clear illustration of the process. By understanding the underlying principles, such as the distributive property and the importance of maintaining balance, we can confidently tackle a wide range of equations. Remember, the key is to break down the problem into manageable steps, applying algebraic operations strategically to isolate the variable. With practice and a solid understanding of these techniques, you'll become a proficient equation solver.