Solving Linear Equations A Step-by-Step Guide

Introduction

In the realm of mathematics, solving equations is a fundamental skill. Equations are mathematical statements that assert the equality of two expressions. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. In this comprehensive guide, we will delve into the process of solving the linear equation -10x + 1 + 7x = 37. Linear equations are algebraic equations where the highest power of the variable is 1. Mastering the techniques to solve linear equations is crucial for success in algebra and beyond. This article will provide a step-by-step approach, ensuring clarity and understanding for learners of all levels. Let's embark on this mathematical journey and unravel the solution to this equation together.

Understanding the Equation: -10x + 1 + 7x = 37

Before diving into the solution, let's break down the equation -10x + 1 + 7x = 37. This is a linear equation in one variable, 'x'. Linear equations are characterized by their straight-line graph representation and the fact that the variable is raised to the power of 1. The equation consists of two sides separated by an equals sign (=). The left side, -10x + 1 + 7x, is an expression involving the variable 'x' and constant terms. The right side, 37, is a constant value. Our objective is to isolate 'x' on one side of the equation to determine its value. To achieve this, we will employ algebraic manipulations, ensuring that we perform the same operations on both sides of the equation to maintain the balance. Understanding the structure of the equation is the first step towards finding its solution. This involves recognizing the terms, the coefficients, and the constant values, all of which play a role in the solving process.

Step 1: Combine Like Terms

The first crucial step in solving the equation -10x + 1 + 7x = 37 is to combine like terms. Like terms are those that have the same variable raised to the same power. In this equation, we have two terms involving 'x': -10x and +7x. Combining these terms involves adding their coefficients. The coefficient of a term is the numerical factor that multiplies the variable. In this case, the coefficients are -10 and 7. Adding these coefficients gives us -10 + 7 = -3. Therefore, the combined term is -3x. Now, let's rewrite the equation with the like terms combined. The original equation is -10x + 1 + 7x = 37. Combining -10x and 7x, we get -3x. So, the equation becomes -3x + 1 = 37. This simplification makes the equation easier to work with, bringing us closer to isolating the variable 'x'. Combining like terms is a fundamental algebraic technique that simplifies equations and paves the way for further steps in the solving process.

Step 2: Isolate the Variable Term

After combining like terms, the next step in solving our equation -3x + 1 = 37 is to isolate the variable term. The variable term is the term that contains the variable, which in this case is -3x. To isolate -3x, we need to eliminate any other terms on the same side of the equation. In our equation, we have a constant term, +1, on the left side along with -3x. To eliminate +1, we perform the inverse operation, which is subtraction. We subtract 1 from both sides of the equation. This ensures that we maintain the balance of the equation, as whatever we do to one side, we must do to the other. Subtracting 1 from both sides gives us: -3x + 1 - 1 = 37 - 1. Simplifying this, we get -3x = 36. Now, the variable term, -3x, is isolated on the left side of the equation. This is a significant step forward in solving for 'x', as we have successfully removed the constant term from the same side as the variable. The next step will involve isolating 'x' itself.

Step 3: Solve for x

Now that we have isolated the variable term, -3x = 36, the final step in solving the equation is to solve for 'x'. This involves getting 'x' by itself on one side of the equation. Currently, 'x' is being multiplied by -3. To isolate 'x', we need to perform the inverse operation, which is division. We will divide both sides of the equation by -3. Dividing both sides by -3 gives us: (-3x) / -3 = 36 / -3. On the left side, -3 divided by -3 cancels out, leaving us with just 'x'. On the right side, 36 divided by -3 is -12. Therefore, the equation simplifies to x = -12. This is the solution to the equation -10x + 1 + 7x = 37. We have successfully found the value of 'x' that makes the equation true. To verify our solution, we can substitute x = -12 back into the original equation and check if both sides are equal. Solving for 'x' is the ultimate goal in solving an equation, and we have achieved it through a series of algebraic manipulations.

Step 4: Verify the Solution

After solving an equation, it's always a good practice to verify the solution. This ensures that the value we found for the variable indeed satisfies the original equation. In our case, we found that x = -12 is the solution to the equation -10x + 1 + 7x = 37. To verify this, we substitute x = -12 back into the original equation and check if both sides are equal. Substituting x = -12 into the left side of the equation gives us: -10(-12) + 1 + 7(-12). Let's simplify this expression: -10(-12) = 120, 7(-12) = -84. So, the expression becomes 120 + 1 - 84. Adding these numbers, we get 121 - 84 = 37. The left side of the equation simplifies to 37, which is the same as the right side of the original equation. This confirms that our solution, x = -12, is correct. Verification is a crucial step in the problem-solving process, as it helps to catch any errors and build confidence in the solution. By substituting the solution back into the original equation, we can ensure that the equation holds true.

Conclusion

In conclusion, we have successfully navigated the process of solving the linear equation -10x + 1 + 7x = 37. Through a step-by-step approach, we combined like terms, isolated the variable term, solved for x, and verified the solution. The key steps involved combining like terms to simplify the equation, isolating the variable term by performing inverse operations, solving for x by dividing both sides by the coefficient of x, and verifying the solution by substituting it back into the original equation. The solution we found is x = -12. This comprehensive guide has provided a clear and concise method for solving linear equations, equipping learners with the necessary skills to tackle similar problems. Mastering these techniques is essential for success in algebra and beyond. Remember, practice is key to honing your problem-solving abilities. By working through various examples, you can solidify your understanding and become more confident in your mathematical endeavors. Solving equations is a fundamental skill that opens doors to more advanced mathematical concepts, and with a solid foundation, you can excel in your mathematical journey.