Solving The Equation 8 - 3x = 20 A Step-by-Step Guide

Solving equations is a fundamental skill in mathematics, and understanding the steps involved is crucial for success in algebra and beyond. In this comprehensive guide, we will break down the process of solving a given equation step-by-step, providing clear explanations and justifications for each step. We will focus on the example equation $8 - 3x = 20$, demonstrating how to isolate the variable and find the solution. This guide will not only help you understand the mechanics of solving equations but also the underlying principles that make it work. Mastering these concepts will empower you to tackle more complex problems with confidence. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide provides a solid foundation for your mathematical journey.

Understanding the Equation: $8 - 3x = 20$

Before diving into the steps, let's first understand the equation we're working with: $8 - 3x = 20$. This equation states that the expression $8 - 3x$ is equal to 20. Our goal is to find the value of the variable $x$ that makes this statement true. In essence, we want to isolate $x$ on one side of the equation. This involves performing operations on both sides of the equation to maintain the equality while gradually simplifying the expression until we have $x$ by itself. Understanding the structure of the equation is the first step toward solving it. We have a constant term (8), a variable term (-3x), and a constant on the other side of the equation (20). By strategically applying mathematical operations, we can unravel the equation and reveal the value of $x$.

Step 1: Applying the Subtraction Property of Equality

The first step in solving the equation $8 - 3x = 20$ involves using the subtraction property of equality. This property states that if we subtract the same value from both sides of an equation, the equation remains balanced. In this case, we want to eliminate the constant term (+8) from the left side of the equation. To do this, we subtract 8 from both sides:

$8 - 3x - 8 = 20 - 8$

This step is crucial because it isolates the term containing the variable ($x$) on the left side. By subtracting 8, we are essentially undoing the addition of 8, bringing us closer to isolating $x$. The subtraction property of equality is a fundamental principle in algebra, ensuring that any operation performed maintains the equation's balance. Without this property, we couldn't confidently manipulate equations to solve for unknowns. This step demonstrates the importance of maintaining balance and using inverse operations to simplify equations.

Step 2: Simplifying the Equation

After applying the subtraction property of equality, we simplify both sides of the equation. On the left side, $8 - 3x - 8$ simplifies to $-3x$, as the +8 and -8 cancel each other out. On the right side, $20 - 8$ simplifies to 12. This gives us the simplified equation:

$-3x = 12$

This simplified equation is much easier to work with than the original. We have reduced the equation to a single term containing the variable on one side and a constant on the other. This step highlights the power of simplification in mathematics. By combining like terms and performing basic arithmetic, we can transform a complex equation into a more manageable form. The simplified equation $-3x = 12$ sets the stage for the next step, where we will isolate $x$ completely. This step is a testament to the importance of simplifying expressions to make problem-solving more efficient.

Step 3: The Division Property of Equality

To isolate the variable $x$, we need to undo the multiplication by -3. This is where the division property of equality comes into play. The division property of equality states that if we divide both sides of an equation by the same non-zero value, the equation remains balanced. In this case, we divide both sides of the equation $-3x = 12$ by -3:

rac{-3x}{-3} = rac{12}{-3}

This step is critical because it directly addresses the coefficient of $x$. By dividing by -3, we are effectively canceling out the -3 that is multiplying $x$. The division property of equality is another cornerstone of algebraic manipulation, allowing us to isolate variables and solve for their values. This step demonstrates the inverse relationship between multiplication and division and how we can use this relationship to our advantage in solving equations. This step is essential for getting $x$ by itself and finding the solution.

Step 4: Determining the Solution

After dividing both sides of the equation by -3, we simplify to find the value of $x$. On the left side, rac{-3x}{-3} simplifies to $x$. On the right side, rac{12}{-3} simplifies to -4. Therefore, the solution to the equation is:

$x = -4$

This final step reveals the value of $x$ that satisfies the original equation. We have successfully isolated $x$ and found its value. The solution $x = -4$ means that if we substitute -4 for $x$ in the original equation $8 - 3x = 20$, the equation will be true. This step is the culmination of all the previous steps, demonstrating the power of algebraic manipulation to solve for unknowns. This is a satisfying conclusion to the problem-solving process, showing that by systematically applying the properties of equality, we can find the solution to any equation. Verifying this solution by substituting it back into the original equation is a good practice to ensure accuracy.

Verifying the Solution

To ensure our solution $x = -4$ is correct, we substitute it back into the original equation $8 - 3x = 20$:

$8 - 3(-4) = 20$

Now, we simplify the left side of the equation:

$8 + 12 = 20$

$20 = 20$

Since the left side equals the right side, our solution $x = -4$ is correct. This verification step is a crucial part of the problem-solving process. It provides a check on our work and ensures that we have not made any errors along the way. By substituting the solution back into the original equation, we can confirm that it satisfies the equation's conditions. This step not only gives us confidence in our answer but also reinforces the understanding of what it means to solve an equation. Verification is a valuable habit to develop in mathematics, as it promotes accuracy and deepens understanding.

Conclusion

In conclusion, solving equations is a systematic process that involves applying the properties of equality to isolate the variable. We have demonstrated this process step-by-step using the example equation $8 - 3x = 20$. By understanding and applying the subtraction and division properties of equality, we were able to find the solution $x = -4$. This guide has not only shown the mechanics of solving equations but also the underlying principles that make it work. Mastering these concepts is essential for success in algebra and beyond. The ability to solve equations is a fundamental skill that opens doors to more advanced mathematical concepts and real-world applications. By following these steps and practicing regularly, you can build your confidence and proficiency in solving equations of all kinds. Remember, mathematics is a journey, and each solved equation is a step forward.