Solving $-4k + 2(5k - 6) = -3k - 39$ A Step-by-Step Guide
In this article, we will delve into the process of solving the linear equation -4k + 2(5k - 6) = -3k - 39. Linear equations are fundamental in mathematics and appear in various real-world applications. Understanding how to solve them is a crucial skill. This guide will provide a detailed, step-by-step explanation to help you grasp the solution method effectively. By the end of this article, you'll have a solid understanding of how to tackle similar linear equations with confidence.
Understanding the Equation
Before we dive into the solution, let's first understand the given equation: -4k + 2(5k - 6) = -3k - 39. This is a linear equation in one variable, k. Our goal is to find the value of k that makes the equation true. To do this, we will use algebraic manipulations to isolate k on one side of the equation. The equation involves terms with k, constant terms, and parentheses. We need to follow the correct order of operations to simplify the equation and solve for k. The process involves distributing, combining like terms, and performing inverse operations to isolate the variable. Let's break down each step to make the solution clear and easy to follow. Understanding the structure of the equation is the first step towards successfully solving it. Linear equations are prevalent in many areas of mathematics and real-world applications, making it essential to have a strong grasp of the solving process. By understanding the components of the equation, we can apply the appropriate techniques to find the solution.
Step 1: Distribute the 2
The first step in solving the equation -4k + 2(5k - 6) = -3k - 39 is to eliminate the parentheses. We do this by distributing the 2 across the terms inside the parentheses. This means we multiply 2 by both 5k and -6. Performing this distribution gives us: 2 * 5k = 10k and 2 * -6 = -12. So, the equation now becomes: -4k + 10k - 12 = -3k - 39. Distributing correctly is a crucial step because it simplifies the equation and allows us to combine like terms in the next step. Without proper distribution, we cannot proceed to simplify the equation further. This step follows the order of operations (PEMDAS/BODMAS), which prioritizes multiplication over addition and subtraction. The distribution step effectively removes the parentheses, making the equation easier to work with. By distributing, we maintain the equation's balance while transforming it into a more manageable form. This step is a cornerstone of solving many algebraic equations and is an essential skill to master.
Step 2: Combine Like Terms on the Left Side
After distributing, our equation is -4k + 10k - 12 = -3k - 39. Now, we need to combine the like terms on the left side of the equation. Like terms are terms that have the same variable raised to the same power. In this case, -4k and 10k are like terms. To combine them, we add their coefficients: -4 + 10 = 6. So, -4k + 10k becomes 6k. The equation now simplifies to: 6k - 12 = -3k - 39. Combining like terms simplifies the equation further, making it easier to isolate the variable k. This step reduces the number of terms in the equation, which helps in the subsequent steps of solving for k. By combining like terms, we are essentially consolidating the terms involving k into a single term. This process ensures that the equation remains balanced and accurate. This step is an important part of the simplification process and is used extensively in algebra. Simplifying the equation by combining like terms allows us to focus on isolating the variable and finding its value more efficiently.
Step 3: Add 3k to Both Sides
Our equation is currently 6k - 12 = -3k - 39. To isolate the variable k, we need to get all the terms with k on one side of the equation. A good approach is to add 3k to both sides of the equation. This will eliminate the -3k term on the right side. Adding 3k to both sides, we get: 6k + 3k - 12 = -3k + 3k - 39. Simplifying this, we have: 9k - 12 = -39. Adding the same term to both sides of the equation maintains the balance, which is a fundamental principle in solving equations. This step moves us closer to isolating k by consolidating the k terms on the left side. By adding 3k, we have successfully removed the variable term from the right side, simplifying the equation further. This is a common technique in solving algebraic equations and is crucial for isolating the variable. The result is a more streamlined equation that is easier to work with in the next steps.
Step 4: Add 12 to Both Sides
Now we have the equation 9k - 12 = -39. Our next goal is to isolate the term with k by eliminating the constant term on the left side. We can do this by adding 12 to both sides of the equation. Adding 12 to both sides, we get: 9k - 12 + 12 = -39 + 12. This simplifies to: 9k = -27. Adding the same value to both sides maintains the equation's balance and helps us isolate the term with the variable. This step eliminates the constant term on the left side, bringing us closer to solving for k. By adding 12, we are effectively undoing the subtraction of 12, which was part of the original equation. This step is a common algebraic manipulation used to simplify equations and isolate variables. The resulting equation is much simpler and makes it easier to find the value of k in the next step.
Step 5: Divide Both Sides by 9
We've reached the equation 9k = -27. To finally solve for k, we need to isolate k completely. Since k is being multiplied by 9, we can undo this multiplication by dividing both sides of the equation by 9. Dividing both sides by 9, we get: (9k) / 9 = -27 / 9. This simplifies to: k = -3. Dividing both sides by the same number keeps the equation balanced and isolates k. This is the final step in solving the equation, giving us the value of k. By dividing, we are performing the inverse operation of multiplication, which allows us to find the value of the variable. The solution k = -3 means that if we substitute -3 for k in the original equation, the equation will be true. This step completes the process of solving for k and provides the solution to the equation.
Solution: k = -3
After following all the steps, we have found the solution to the equation -4k + 2(5k - 6) = -3k - 39. The solution is k = -3. This means that when we substitute -3 for k in the original equation, both sides of the equation will be equal. Let's verify this by substituting k = -3 into the original equation: -4(-3) + 2(5(-3) - 6) = -3(-3) - 39. Simplifying the left side: 12 + 2(-15 - 6) = 12 + 2(-21) = 12 - 42 = -30. Simplifying the right side: -3(-3) - 39 = 9 - 39 = -30. Since both sides equal -30, our solution k = -3 is correct. Verifying the solution is an important step to ensure accuracy and confirm that we have solved the equation correctly. The solution k = -3 is the only value that makes the equation true. This solution provides a concrete answer to the problem and demonstrates the effectiveness of the step-by-step method we used to solve the equation. Understanding how to solve linear equations like this one is crucial in mathematics and its applications.
Conclusion
In this detailed guide, we have walked through the process of solving the linear equation -4k + 2(5k - 6) = -3k - 39. We started by understanding the equation, then we followed a step-by-step approach: distributing, combining like terms, adding terms to both sides, and finally dividing to isolate the variable k. The solution we found is k = -3, which we verified by substituting it back into the original equation. Mastering the process of solving linear equations is essential in mathematics, as it forms the foundation for more complex algebraic concepts. By breaking down the problem into manageable steps, we can tackle even seemingly complicated equations with confidence. Each step, from distribution to combining like terms and isolating the variable, plays a crucial role in reaching the correct solution. This process not only helps in solving equations but also enhances problem-solving skills in general. Linear equations are prevalent in various real-world scenarios, making the ability to solve them a valuable skill. Understanding and practicing these techniques will empower you to approach and solve a wide range of mathematical problems effectively. The methodical approach demonstrated here can be applied to other linear equations, reinforcing the importance of understanding each step in the solving process. Remember, practice is key to mastering these skills, so try solving similar equations to solidify your understanding.
