Solving Linear Equations: A Step-by-Step Guide

Have you ever felt lost when faced with a seemingly complex equation? Don't worry, you're not alone! Many students find algebra challenging, but with the right approach, solving equations can become a breeze. In this article, we'll break down the process of solving linear equations, using the example -8d - 5d + 7d = 72 as our guide. We'll walk through each step in detail, ensuring you understand the underlying concepts and can confidently tackle similar problems in the future. So, let's dive in and unlock the secrets of equation-solving!

Understanding the Basics of Linear Equations

Before we jump into solving the equation, let's establish a solid foundation by understanding what a linear equation is and the key principles involved in solving them. Linear equations are algebraic equations where the highest power of the variable is 1. They represent a straight line when graphed, hence the name "linear." Our example equation, -8d - 5d + 7d = 72, fits this definition perfectly. The variable 'd' has a power of 1, making it a linear equation.

The primary goal when solving any equation is to isolate the variable on one side of the equation. This means we want to manipulate the equation until we have 'd' (in our case) by itself on either the left or right side. To achieve this, we rely on the fundamental principle of equality: whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the balance. Think of an equation as a perfectly balanced scale; if you add or remove weight from one side, you must do the same on the other to keep it balanced. This principle is the cornerstone of equation-solving, allowing us to add, subtract, multiply, or divide both sides without changing the equation's solution.

Another crucial concept is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). While PEMDAS is essential for simplifying expressions, when solving equations, we often work in reverse, undoing operations to isolate the variable. This means we typically deal with addition and subtraction before multiplication and division. By understanding these basic principles – the definition of a linear equation, the principle of equality, and the order of operations – you'll be well-equipped to tackle a wide range of equation-solving challenges. Remember, practice makes perfect, so the more you apply these concepts, the more confident you'll become!

Step 1: Combining Like Terms

Our first step in solving the equation -8d - 5d + 7d = 72 is to simplify the left side by combining like terms. Like terms are those that contain the same variable raised to the same power. In this equation, we have three terms containing the variable 'd': -8d, -5d, and 7d. To combine them, we simply add or subtract their coefficients (the numbers in front of the variable). This is a crucial step in simplifying the equation and making it easier to solve. By reducing the number of terms, we streamline the process and minimize the chances of errors.

Let's start by combining -8d and -5d. Both terms have negative coefficients, so we add their absolute values and keep the negative sign. This gives us -8 + (-5) = -13, so -8d - 5d simplifies to -13d. Now our equation looks like this: -13d + 7d = 72. We've successfully combined two of the like terms, making the equation slightly less cluttered. Next, we combine -13d with the remaining term, 7d. Here, we have a negative term and a positive term. To combine them, we subtract the smaller absolute value from the larger absolute value and use the sign of the term with the larger absolute value. In this case, we have -13 + 7. The absolute value of -13 is 13, and the absolute value of 7 is 7. Subtracting 7 from 13 gives us 6. Since -13 has a larger absolute value and is negative, the result is -6. Therefore, -13d + 7d simplifies to -6d.

Now, our equation is significantly simpler: -6d = 72. By combining like terms, we've reduced the equation to a more manageable form. This step is essential for isolating the variable and moving closer to the solution. Remember, combining like terms is like tidying up a messy room – it makes everything easier to find and work with. With this simplified equation, we're ready to move on to the next step, where we'll isolate the variable 'd' and finally solve for its value. Keep practicing combining like terms, and you'll find it becomes second nature, making equation-solving a much smoother process.

Step 2: Isolating the Variable

Now that we've simplified the equation to -6d = 72, our next crucial step is isolating the variable 'd'. This means we need to get 'd' by itself on one side of the equation. Currently, 'd' is being multiplied by -6. To undo this multiplication and isolate 'd', we'll use the inverse operation: division. Remember the principle of equality we discussed earlier? It's essential here. Whatever we do to one side of the equation, we must do to the other to maintain the balance.

Since 'd' is being multiplied by -6, we'll divide both sides of the equation by -6. This will effectively "cancel out" the -6 on the left side, leaving 'd' by itself. When we divide the left side (-6d) by -6, the -6 in the numerator and the -6 in the denominator cancel each other out, resulting in just 'd'. This is exactly what we want – 'd' isolated! Now, we need to perform the same operation on the right side of the equation. We divide 72 by -6. A positive number divided by a negative number results in a negative number. 72 divided by 6 is 12, so 72 divided by -6 is -12.

After performing the division on both sides, our equation now looks like this: d = -12. We've successfully isolated the variable! This is the solution to our equation. It means that if we substitute -12 for 'd' in the original equation, the equation will hold true. Isolating the variable is a fundamental technique in solving equations. It's like peeling away the layers of an onion to reveal the core. By understanding inverse operations and consistently applying the principle of equality, you can isolate any variable and find the solution to the equation. This step empowers you to take control of the equation and systematically unravel its mystery. So, remember, the key is to identify the operation being performed on the variable and then use the inverse operation to isolate it. With practice, you'll become a pro at isolating variables and solving equations with confidence.

Step 3: Verifying the Solution

We've arrived at a potential solution for our equation -8d - 5d + 7d = 72: we found that d = -12. However, it's crucial to verify our solution to ensure we haven't made any errors along the way. This step is like double-checking your work on an important assignment – it gives you confidence in your answer and helps you catch any mistakes. Verifying the solution involves substituting the value we found for 'd' back into the original equation and checking if both sides of the equation are equal. This process confirms that our solution satisfies the equation and is indeed correct.

To verify, we'll replace 'd' with -12 in the original equation: -8(-12) - 5(-12) + 7(-12) = 72. Now, we need to simplify the left side of the equation following the order of operations. First, we perform the multiplications: -8 multiplied by -12 is 96 (a negative times a negative is a positive). -5 multiplied by -12 is 60. And 7 multiplied by -12 is -84. So, our equation now looks like this: 96 + 60 - 84 = 72. Next, we perform the addition and subtraction from left to right. 96 plus 60 is 156. So, we have 156 - 84 = 72. Finally, we subtract 84 from 156, which gives us 72. Now, our equation reads 72 = 72.

Since the left side of the equation (72) is equal to the right side of the equation (72), our solution d = -12 is correct! This verification step provides us with the assurance that we've accurately solved the equation. Verifying your solution is a vital habit to cultivate when solving equations. It not only confirms your answer but also reinforces your understanding of the equation-solving process. By substituting your solution back into the original equation, you're essentially testing whether the value you found makes the equation true. If it does, you've solved the equation correctly. If not, you know there's an error somewhere, and you can go back and review your steps. So, always remember to verify your solutions – it's the final piece of the puzzle in mastering equation-solving.

Conclusion

In this article, we've embarked on a journey to solve the linear equation -8d - 5d + 7d = 72. We've broken down the process into three manageable steps: combining like terms, isolating the variable, and verifying the solution. By combining like terms, we simplified the equation to -6d = 72. Then, by isolating the variable 'd', we found that d = -12. Finally, we verified our solution by substituting -12 back into the original equation, confirming that it is indeed the correct answer. This step-by-step approach not only helps you solve the specific equation at hand but also equips you with a powerful framework for tackling a wide range of linear equations.

Solving equations is a fundamental skill in mathematics and has applications in various fields, from science and engineering to finance and economics. Mastering this skill opens doors to more advanced mathematical concepts and problem-solving opportunities. Remember, the key to success in equation-solving is practice. The more you work through different types of equations, the more confident and proficient you'll become. Don't be discouraged by challenges; view them as opportunities to learn and grow. Each equation you solve strengthens your understanding and builds your mathematical toolkit.

So, continue to practice, explore different equation-solving techniques, and embrace the challenges. With dedication and perseverance, you'll become a master of equation-solving, unlocking the power of mathematics to solve problems in the real world. For further learning and exploration, you can visit resources like Khan Academy, which offers a wealth of free educational materials on algebra and other mathematical topics. Keep learning, keep practicing, and keep solving!