Solving Linear Equations: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of linear equations and tackling a specific problem. We'll break down how to solve the equation 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1 step by step. Don't worry, it's not as scary as it looks! We’ll walk through each stage, making sure you understand the logic behind every move. Linear equations are a foundational concept in mathematics, appearing in various fields from basic algebra to advanced calculus. Mastering them is crucial for anyone looking to excel in math or related disciplines. This guide not only provides a solution to the given equation but also offers a broader understanding of the strategies involved in solving linear equations, preparing you for more complex problems in the future. So, let's put on our math hats and get started! Remember, practice makes perfect, and understanding the underlying principles is key to success in mathematics. Stick with me, and you'll be solving linear equations like a pro in no time!
1. Understand the Linear Equation
First, let's understand what a linear equation is. Simply put, it's an equation where the highest power of the variable (in our case, 'j') is 1. Our equation, 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1, fits this definition perfectly. Before we jump into solving, let's identify the key components of a linear equation. Linear equations typically involve variables, coefficients, constants, and an equals sign, which indicates that the expressions on both sides of the equation are equal. The goal of solving a linear equation is to isolate the variable on one side of the equation, allowing us to determine its value. This involves performing algebraic operations such as addition, subtraction, multiplication, and division while maintaining the balance of the equation. Understanding these fundamentals is essential for tackling more complex linear equations and systems of equations. Linear equations are not just abstract mathematical concepts; they have practical applications in various real-world scenarios, such as modeling relationships between quantities, solving problems in physics and engineering, and making predictions in economics and finance. Therefore, mastering linear equations provides a valuable toolset for problem-solving in diverse fields.
2. Combine Like Terms
The first step in solving our equation is to combine like terms. This means grouping together the 'j' terms and the constant terms on each side of the equation. Look at the left side: 2.25 - 11j - 7.75 + 1.5j. We have two constant terms (2.25 and -7.75) and two 'j' terms (-11j and 1.5j). Let's combine them. Combining like terms simplifies the equation, making it easier to isolate the variable and solve for its value. This step involves adding or subtracting the coefficients of the variable terms and the constant terms separately. By grouping similar terms together, we reduce the complexity of the equation and reveal the underlying structure, which aids in the subsequent steps of the solution process. This technique is not only applicable to solving linear equations but also to simplifying algebraic expressions in general, making it a fundamental skill in algebra. Understanding the concept of like terms and how to combine them efficiently is crucial for mastering algebraic manipulations and solving a wide range of mathematical problems. Practice combining like terms in various contexts to build proficiency and confidence in algebraic manipulations.
Combining Constant Terms
Let's start with the constants: 2.25 - 7.75 = -5.5. So, the constant part simplifies to -5.5. Combining constants is a fundamental arithmetic operation that simplifies the equation by reducing multiple numerical terms into a single constant. This step helps to clarify the equation's structure and prepares it for further algebraic manipulations. It's important to pay close attention to the signs of the constants, as both positive and negative values need to be correctly accounted for. The result of combining constants is a single numerical value that represents the net effect of all the constant terms in the equation. This process not only simplifies the equation but also allows for a more intuitive understanding of the relationships between different parts of the equation. Mastering the combination of constants is essential for solving linear equations and performing other mathematical operations involving numerical values.
Combining 'j' Terms
Now, let's combine the 'j' terms: -11j + 1.5j = -9.5j. So, the 'j' part simplifies to -9.5j. Combining variable terms, like the 'j' terms in this case, is a crucial step in simplifying algebraic expressions and solving equations. This process involves adding or subtracting the coefficients of the variable terms while keeping the variable itself unchanged. It's essential to ensure that only like terms, which have the same variable raised to the same power, are combined. This step reduces the number of terms in the equation, making it easier to isolate the variable and find its value. Combining variable terms also helps in recognizing patterns and relationships within the equation, which can lead to a deeper understanding of the underlying mathematical structure. Proficiency in combining variable terms is essential for success in algebra and other advanced mathematical topics.
3. Rewrite the Equation
After combining like terms on the left side, our equation looks like this: -5.5 - 9.5j = 0.5j - 1. See how much simpler it is now? Rewriting the equation after each simplification step is a good practice as it helps to keep track of the progress and ensures that the equation remains balanced. This step may seem trivial, but it plays a crucial role in maintaining clarity and accuracy throughout the solution process. By rewriting the equation, we reinforce our understanding of the transformations that have been made and prepare for the next stage of simplification. This practice is particularly helpful when dealing with more complex equations that require multiple steps to solve. It also allows for easy verification of each step, ensuring that no errors have been introduced along the way. Rewriting the equation is a simple yet effective technique for improving problem-solving efficiency and accuracy in algebra.
4. Isolate the 'j' Terms
Next, we want to isolate the 'j' terms on one side of the equation. Let's move the 0.5j term from the right side to the left side. To do this, we subtract 0.5j from both sides of the equation: -5.5 - 9.5j - 0.5j = 0.5j - 1 - 0.5j. Isolating the variable terms is a fundamental step in solving equations, as it brings all terms containing the variable to one side of the equation, allowing us to eventually solve for the variable's value. This step typically involves adding or subtracting terms from both sides of the equation to maintain balance while moving the variable terms to the desired side. The goal is to simplify the equation so that only the variable term and constant terms remain, setting the stage for further algebraic manipulations. Proficiency in isolating variable terms is essential for solving a wide range of equations and problems in mathematics.
Simplify After Moving 'j' Terms
This simplifies to -5.5 - 10j = -1. Notice how -9.5j - 0.5j becomes -10j. Remember, we always perform the same operation on both sides of the equation to maintain balance. Simplifying the equation after moving variable terms involves combining like terms and reducing the complexity of the expression. This step is crucial for maintaining clarity and accuracy throughout the solution process. By simplifying the equation, we make it easier to identify the next steps required to isolate the variable and solve for its value. This process may involve combining constant terms, variable terms, or both, depending on the specific equation. The goal is to create a simplified equation that is easier to work with and understand.
5. Isolate the Constant Terms
Now, let's isolate the constant terms. We'll move the -5.5 from the left side to the right side. To do this, we add 5.5 to both sides: -5.5 - 10j + 5.5 = -1 + 5.5. Isolating constant terms is a crucial step in solving equations, as it involves moving all numerical values to one side of the equation, leaving only the variable terms on the other side. This step typically involves adding or subtracting constants from both sides of the equation to maintain balance. The goal is to simplify the equation so that only the variable term and a single constant remain, making it easier to solve for the variable's value.
Simplify Again!
This simplifies to -10j = 4.5. We're getting closer! Remember to double-check your arithmetic as you go. Careful arithmetic is essential for maintaining accuracy in solving equations. Errors in calculations can lead to incorrect solutions, so it's important to double-check each step and ensure that all operations are performed correctly. This includes paying attention to signs, fractions, decimals, and other mathematical details. By practicing careful arithmetic, you can minimize the risk of errors and improve your problem-solving skills.
6. Solve for 'j'
Finally, to solve for 'j', we need to get it all by itself. We have -10j = 4.5. This means -10 times j equals 4.5. To undo the multiplication, we divide both sides by -10: (-10j) / -10 = 4.5 / -10. Solving for the variable is the ultimate goal of equation solving, as it involves determining the value of the variable that satisfies the equation. This step typically involves performing inverse operations to isolate the variable on one side of the equation. It's crucial to ensure that the operations are performed correctly and that all steps are followed logically to arrive at the correct solution.
The Final Step
This gives us j = -0.45. Woohoo! We've solved the equation! The final step in solving an equation involves presenting the solution clearly and verifying that it satisfies the original equation. This ensures that the solution is correct and that all steps have been followed accurately. It's also helpful to check the solution by substituting it back into the original equation and confirming that both sides of the equation are equal.
7. Check Your Answer (Optional but Recommended)
It's always a good idea to check your answer. Plug j = -0.45 back into the original equation: 2.25 - 11(-0.45) - 7.75 + 1.5(-0.45) = 0.5(-0.45) - 1. Checking the answer is a crucial step in the problem-solving process, as it verifies the correctness of the solution and ensures that no errors have been made along the way. This step involves substituting the solution back into the original equation and confirming that both sides of the equation are equal. If the equation holds true, then the solution is correct; otherwise, it indicates that an error has occurred, and the solution process needs to be reviewed. Checking the answer not only confirms the correctness of the solution but also reinforces understanding of the problem and the steps involved in solving it.
Verify the Solution
Let's do the math: 2.25 + 4.95 - 7.75 - 0.675 = -0.225 - 1. This simplifies to -1.225 = -1.225. Yay! It checks out! Verifying the solution is the final step in the problem-solving process, as it confirms that the calculated value satisfies the original equation. This step involves substituting the solution back into the equation and simplifying both sides to see if they are equal. If both sides are equal, then the solution is correct; otherwise, an error has occurred, and the solution process needs to be reviewed. Verifying the solution not only ensures the accuracy of the answer but also enhances understanding of the equation and the relationships between its components.
Conclusion
So, guys, we've successfully solved the linear equation 2.25 - 11j - 7.75 + 1.5j = 0.5j - 1, and found that j = -0.45. Remember the key steps: combine like terms, isolate the variable terms, isolate the constant terms, and solve for the variable. Keep practicing, and you'll become a pro at solving linear equations! Congratulations on solving the linear equation! This accomplishment demonstrates your understanding of the fundamental principles and techniques involved in algebraic problem-solving. By breaking down the equation into manageable steps and applying the appropriate algebraic operations, you successfully isolated the variable and found its value. This experience enhances your confidence in tackling similar problems and reinforces your mathematical skills. Keep practicing and exploring different types of equations to further develop your problem-solving abilities and mathematical proficiency. Remember, mathematics is a journey of continuous learning and discovery, and each solved problem is a step forward in that journey.