Solving Linear Equations: A Simple Step-by-Step Guide

Welcome, math explorers! Today, we're diving into the world of algebraic equations, specifically focusing on a common type: linear equations. These equations might seem a bit daunting at first glance, but with a little practice and a clear understanding of the steps involved, you'll be solving them like a pro in no time. Our journey today will center around a specific example: the equation $-9n + 10 = 145$. We'll break down the process of solving this equation, starting with the very first, crucial step. Understanding these foundational techniques is absolutely essential for tackling more complex problems down the line, so let's get started on building a solid mathematical foundation together. We aim to make this process as straightforward and intuitive as possible, ensuring that everyone can grasp the concepts, regardless of their current math background. Our goal is not just to solve this one equation, but to equip you with the understanding of why we perform each step, making you a more confident and capable problem-solver.

The First Step in Solving $-9n + 10 = 145$

When you're faced with an equation like $-9n + 10 = 145$, your primary objective is to isolate the variable, which in this case is 'n'. Think of it like trying to get 'n' all by itself on one side of the equals sign. To do this, we need to undo the operations that are being performed on 'n'. Looking at the equation, we see that 'n' is being multiplied by -9, and then 10 is being added to that result. To reverse these operations, we always start with the operations that are outside the term containing the variable. In our equation, the '+ 10' is further away from the '-9n' term than the '-9' multiplication. Therefore, the first step in solving for 'n' in the equation $-9n + 10 = 145$ would be to subtract 10 on both sides. Why subtract 10? Because subtraction is the inverse operation of addition. By subtracting 10 from the left side, we effectively cancel out the '+ 10' that's currently there, bringing us closer to isolating 'n'. It is absolutely crucial to perform the same operation on both sides of the equation to maintain the equality. If you only subtract 10 from one side, the equation would no longer be balanced, and your solution would be incorrect. This principle of performing the same operation on both sides is a cornerstone of solving algebraic equations. It's like a perfectly balanced scale; whatever you do to one side, you must mirror on the other to keep it level. This initial step of removing the constant term adjacent to the variable term is fundamental to simplifying the equation and moving systematically towards finding the value of 'n'. Mastering this first move sets a clear path for subsequent steps, ensuring accuracy and efficiency in your problem-solving.

Why Subtracting 10 is the Key First Move

Let's delve a little deeper into why subtracting 10 is the crucial first step in solving the equation $-9n + 10 = 145$. The goal in algebra is always to isolate the variable. In this equation, 'n' is trapped by two operations: it's multiplied by -9, and then 10 is added to that product. To free 'n', we need to reverse these operations in the correct order. Think about the order of operations (PEMDAS/BODMAS). When solving an equation, we often reverse this order. We deal with addition and subtraction before multiplication and division. In our equation, '+ 10' is an addition operation applied to the term '-9n'. To undo this addition, we use its inverse operation: subtraction. By subtracting 10 from the left side of the equation, we are essentially saying, "Okay, we want to get rid of this '+ 10' here." So, $-9n + 10 - 10$ simplifies to $-9n$. However, the golden rule of algebra is that whatever you do to one side of the equation, you must do to the other to maintain balance. If we just subtracted 10 from the left side, the equation would become $-9n = 145 - 10$, which is unbalanced. Therefore, we perform the same subtraction on the right side: $145 - 10$. This results in $135$. So, after our first step, the equation transforms from $-9n + 10 = 145$ into $-9n = 135$. This new equation is simpler and closer to our goal of finding 'n'. It's a strategic move that effectively eliminates the constant term from the side with the variable, making the next step much more straightforward. This methodical approach, starting with the constant term and using inverse operations, is the bedrock of solving linear equations and ensures that we are always working towards a correct and logical solution.

Moving Forward: The Next Steps

After successfully completing the first step of subtracting 10 from both sides of the equation $-9n + 10 = 145$, we arrived at the simplified equation $-9n = 135$. Now, our variable 'n' is still not completely isolated; it's being multiplied by -9. To undo this multiplication, we need to use the inverse operation, which is division. So, the next step would be to divide both sides of the equation by -9. Again, remember the golden rule: maintain balance! We must perform this division on both the left and the right sides. Dividing the left side, $-9n$, by -9 gives us just 'n' (since -9 f ext`n` / -9 = n). On the right side, we need to divide 135 by -9. This calculation, 135 f ext`div` -9, results in $-15$. Therefore, the solution to our original equation is $n = -15$. It's always a good practice to check your answer by substituting the value of 'n' back into the original equation. Let's do that: $-9(-15) + 10 = 135 + 10 = 145$. Since $145 = 145$, our solution is correct! This systematic approach—identifying the operations, applying inverse operations in the correct order to both sides, and checking the answer—is the universal method for solving linear equations. Each step builds upon the last, moving you closer to the solution in a logical and predictable manner. Understanding these steps empowers you to tackle any linear equation you encounter.

Conclusion: Mastering Linear Equations

We've journeyed through solving the linear equation $-9n + 10 = 145$, highlighting the critical importance of the first step: subtracting 10 from both sides. This initial action, along with the subsequent division by -9, demonstrates the power of using inverse operations to isolate a variable. Remember, the key principles in solving any algebraic equation are to maintain balance by performing the same operation on both sides and to systematically undo the operations affecting the variable. Whether you're dealing with addition, subtraction, multiplication, or division, always apply its inverse operation to both sides of the equation. This methodical approach will serve you well, not just in this specific problem, but in a vast array of mathematical challenges. Practice is your best friend; the more equations you solve, the more intuitive these steps will become. Don't be afraid to tackle different variations of linear equations to build your confidence and proficiency. Understanding the 'why' behind each step makes the 'how' much easier to grasp and remember. Keep practicing, keep questioning, and you'll find yourself becoming increasingly adept at navigating the fascinating landscape of algebra.

For further exploration and practice on solving linear equations, you can visit helpful resources like Khan Academy or Paul's Online Math Notes. These platforms offer a wealth of explanations, examples, and exercises to deepen your understanding.