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In the realm of mathematics, equations serve as the language through which we express relationships between variables and constants. Solving these equations often necessitates isolating a specific variable, a process that involves strategically manipulating the equation to bring the desired variable to one side while keeping all other terms on the opposite side. This article delves into the fundamental principles and techniques involved in isolating variables, using the equation 5.6j - 0.12 = 4 + 1.1j as a guiding example. We will dissect the equation, explore various approaches to isolate the variable j, and ultimately arrive at the correct solution.
Understanding the Equation
Before embarking on the journey of isolating the variable j, it's crucial to have a solid grasp of the equation's structure. The equation 5.6j - 0.12 = 4 + 1.1j is a linear equation, meaning that the variable j appears only to the first power. It consists of two sides, separated by the equals sign (=). The left-hand side (5.6j - 0.12) involves the variable j multiplied by the coefficient 5.6, and a constant term -0.12. The right-hand side (4 + 1.1j) contains a constant term 4 and the variable j multiplied by the coefficient 1.1.
Isolating a variable essentially means rearranging the equation so that the variable stands alone on one side, with all other terms on the opposite side. This is achieved by performing a series of algebraic operations on both sides of the equation, ensuring that the equality remains balanced. The key principle underlying these operations is the golden rule of algebra: whatever operation you perform on one side of the equation, you must perform the same operation on the other side.
Strategies for Isolating the Variable
Several strategies can be employed to isolate the variable j in the equation 5.6j - 0.12 = 4 + 1.1j. Let's explore some of these strategies in detail:
1. Subtracting 5.6 from Both Sides
One approach might be to subtract 5.6 from both sides of the equation. This operation would eliminate the term 5.6j from the left-hand side, potentially simplifying the equation. However, subtracting 5.6 directly wouldn't isolate j effectively, as it would only eliminate the term on the left side without consolidating the j terms on one side. It's a valid algebraic manipulation, but not the most strategic one for isolating j in this case.
2. Subtracting 0.12 from Both Sides
Another tactic could be to subtract 0.12 from both sides of the equation. While this operation would eliminate the constant term -0.12 from the left-hand side, it wouldn't bring us any closer to isolating j. It would merely shift the constant term to the right-hand side, complicating the equation without contributing to our goal. Therefore, subtracting 0.12 isn't the most effective step for isolating j.
3. Subtracting 1.1j from Both Sides
A more promising strategy involves subtracting 1.1j from both sides of the equation. This operation would achieve two crucial objectives: it would eliminate the term 1.1j from the right-hand side, and it would consolidate the j terms on the left-hand side. By subtracting 1.1j from both sides, we effectively move all terms containing j to one side of the equation, bringing us closer to isolating the variable. This is a key step in solving the equation.
4. Subtracting 4j from Both Sides
Subtracting 4j from both sides is another option, but it's not directly present in the given choices. If we were to consider this, it wouldn't be the most efficient first step. While it would help consolidate the j terms, it doesn't align with the immediate goal of simplifying the equation by directly addressing the constant terms or the j terms in a balanced manner. Subtracting 1.1j is a more strategic move in the initial steps of isolating j.
The Optimal Step: Subtracting 1.1j from Both Sides
Based on our analysis, the most effective step to isolate the variable j in the equation 5.6j - 0.12 = 4 + 1.1j is to subtract 1.1j from both sides. This operation lays the groundwork for further simplification and ultimately leads to the isolation of j. Let's demonstrate how this step transforms the equation:
Original equation: 5.6j - 0.12 = 4 + 1.1j
Subtract 1.1j from both sides: 5.6j - 0.12 - 1.1j = 4 + 1.1j - 1.1j
Simplify: 4.5j - 0.12 = 4
As you can see, subtracting 1.1j from both sides has successfully consolidated the j terms on the left-hand side, resulting in a simpler equation. The next steps would involve isolating j further by adding 0.12 to both sides and then dividing by 4.5. However, the critical first step in this process is indeed subtracting 1.1j from both sides.
Why This Step Works
The effectiveness of subtracting 1.1j from both sides stems from the fundamental principles of algebraic manipulation. By performing the same operation on both sides of the equation, we maintain the equality while strategically rearranging the terms. Subtracting 1.1j eliminates the variable term from the right-hand side, allowing us to group all terms containing j on the left-hand side. This consolidation is crucial for isolating j because it allows us to treat all j terms as a single entity, making it easier to isolate the variable.
Additionally, this step aligns with the general strategy of isolating variables in linear equations: first, consolidate the variable terms on one side of the equation, and then consolidate the constant terms on the other side. By subtracting 1.1j, we take a significant step towards consolidating the variable terms, paving the way for further simplification and ultimately the solution of the equation.
Continuing the Isolation Process
While subtracting 1.1j from both sides is the most effective initial step, it's essential to understand that this is only the first step in the isolation process. To fully isolate j, we need to perform additional algebraic operations. After subtracting 1.1j, the equation becomes 4.5j - 0.12 = 4. The next step is to isolate the term containing j by adding 0.12 to both sides:
Add 0.12 to both sides: 4.5j - 0.12 + 0.12 = 4 + 0.12
Simplify: 4.5j = 4.12
Now, we have the equation 4.5j = 4.12. To finally isolate j, we need to divide both sides by the coefficient 4.5:
Divide both sides by 4.5: 4.5j / 4.5 = 4.12 / 4.5
Simplify: j ≈ 0.9156
Therefore, the solution to the equation 5.6j - 0.12 = 4 + 1.1j is approximately j = 0.9156. This example illustrates the importance of understanding the step-by-step process of isolating variables, as well as the significance of each individual operation in achieving the desired result.
Common Pitfalls to Avoid
When isolating variables, it's easy to make mistakes that can lead to incorrect solutions. Here are some common pitfalls to avoid:
1. Performing Operations on Only One Side of the Equation
The golden rule of algebra dictates that any operation performed on one side of the equation must be performed on the other side as well. Failing to do so will disrupt the equality and lead to an incorrect solution. Always ensure that you apply the same operation to both sides of the equation.
2. Incorrectly Combining Like Terms
When simplifying equations, it's crucial to combine like terms correctly. Like terms are terms that have the same variable raised to the same power. For example, 5.6j and 1.1j are like terms, while 5.6j and 0.12 are not. Make sure to combine only like terms and to perform the correct arithmetic operations on their coefficients.
3. Ignoring the Order of Operations
The order of operations (PEMDAS/BODMAS) is essential for correctly simplifying equations. Make sure to perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Ignoring the order of operations can lead to errors in simplification and incorrect solutions.
4. Forgetting to Distribute
When an equation involves parentheses or brackets, it's crucial to distribute any terms outside the parentheses to the terms inside. For example, if you have the expression 2(x + 3), you need to distribute the 2 to both x and 3, resulting in 2x + 6. Forgetting to distribute can lead to incorrect simplification and incorrect solutions.
Conclusion
Isolating variables is a fundamental skill in algebra and a cornerstone of solving equations. In the context of the equation 5.6j - 0.12 = 4 + 1.1j, the most effective initial step is to subtract 1.1j from both sides. This operation consolidates the j terms on one side of the equation, paving the way for further simplification and ultimately the solution. By understanding the principles of algebraic manipulation, employing strategic techniques, and avoiding common pitfalls, you can master the art of isolating variables and confidently solve a wide range of equations.
Remember, isolating variables is not just about following a set of rules; it's about understanding the underlying logic and applying it strategically. Practice and careful attention to detail are key to success in this crucial mathematical skill.
#repair-input-keyword Which step could be used to help isolate the variable in the following equation? #title Correct Step to Isolate Variable j in Equation 5.6j - 0.12 = 4 + 1.1j