Solving The Equation (8x + 4)/5 = X/9 A Step-by-Step Guide

In the realm of mathematics, particularly in algebra, solving equations is a fundamental skill. This article delves into a step-by-step approach to solve the linear equation (8x + 4)/5 = x/9. Understanding how to solve such equations is crucial for various applications in science, engineering, and everyday problem-solving. We will not only provide the solution but also explain the underlying principles and techniques involved. This comprehensive guide aims to equip you with the knowledge and confidence to tackle similar algebraic challenges. Whether you are a student learning the basics or someone looking to refresh your algebraic skills, this detailed explanation will serve as a valuable resource.

Understanding Linear Equations

Before diving into the specifics of the equation (8x + 4)/5 = x/9, let's first grasp the concept of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called “linear” because when plotted on a graph, they form a straight line. The general form of a linear equation is ax + b = 0, where a and b are constants, and x is the variable. Our equation, (8x + 4)/5 = x/9, is a linear equation as it can be rearranged into this general form. Solving a linear equation involves finding the value of the variable (in this case, x) that makes the equation true. This is achieved by isolating the variable on one side of the equation through a series of algebraic manipulations, ensuring that the equality remains balanced. The techniques used to solve linear equations, such as cross-multiplication, distribution, and combining like terms, are essential tools in algebra and form the basis for solving more complex equations and systems of equations. Understanding these fundamentals is key to mastering algebraic problem-solving.

Step-by-Step Solution of (8x + 4)/5 = x/9

Now, let's embark on a detailed, step-by-step journey to solve the equation (8x + 4)/5 = x/9. This process will illustrate the application of various algebraic principles and techniques. Each step is carefully explained to ensure clarity and understanding.

Step 1: Cross-Multiplication

The initial step in solving this equation involves cross-multiplication. This technique is used to eliminate the fractions, making the equation easier to work with. Cross-multiplication is based on the principle that if a/b = c/d, then ad = bc. Applying this to our equation, (8x + 4)/5 = x/9, we multiply the numerator of the left side by the denominator of the right side, and vice versa. This gives us: 9(8x + 4) = 5x. This step effectively removes the fractions, transforming the equation into a more manageable form.

Step 2: Distribution

Next, we apply the distributive property to the left side of the equation. The distributive property states that a(b + c) = ab + ac. Applying this to 9(8x + 4), we multiply 9 by both terms inside the parentheses: 9 * 8x + 9 * 4. This results in 72x + 36. So, our equation now looks like this: 72x + 36 = 5x. Distribution is a crucial step in simplifying equations that involve parentheses or brackets.

Step 3: Isolating the Variable

The next critical step is to isolate the variable x on one side of the equation. This involves moving all terms containing x to one side and constant terms to the other side. To do this, we subtract 5x from both sides of the equation: 72x + 36 - 5x = 5x - 5x. This simplifies to 67x + 36 = 0. The goal here is to group the x terms together, making it easier to solve for x. This step is a fundamental technique in solving algebraic equations.

Step 4: Isolating the Constant

Following the isolation of the variable terms, we now isolate the constant term. We do this by subtracting 36 from both sides of the equation: 67x + 36 - 36 = 0 - 36. This simplifies to 67x = -36. By isolating the constant term, we bring the equation closer to the final solution, where x is by itself on one side of the equation. This step is a crucial part of the process of solving for the unknown variable.

Step 5: Solving for x

Finally, to solve for x, we divide both sides of the equation by the coefficient of x, which is 67. This gives us: 67x / 67 = -36 / 67. This simplifies to x = -36/67. Thus, we have found the solution to the equation. The value of x that satisfies the equation (8x + 4)/5 = x/9 is -36/67. This final step completes the process of isolating x and determining its value.

Verification of the Solution

To ensure the accuracy of our solution, it is essential to verify the solution. This involves substituting the value of x we found back into the original equation and checking if both sides of the equation are equal. This process confirms that our solution is correct and that no errors were made during the solving process.

Substituting x = -36/67 into the Original Equation

We substitute x = -36/67 into the original equation (8x + 4)/5 = x/9. This gives us (8(-36/67) + 4)/5 = (-36/67)/9. Now, we simplify each side of the equation separately.

Simplifying the Left Side

First, we simplify the numerator of the left side: 8 * (-36/67) = -288/67. Then, we add 4 to this result: -288/67 + 4. To add these, we need a common denominator, which is 67. So, we rewrite 4 as 268/67. Now we have -288/67 + 268/67 = -20/67. Next, we divide this result by 5: (-20/67) / 5. Dividing by 5 is the same as multiplying by 1/5, so we have (-20/67) * (1/5) = -20/335. Simplifying this fraction by dividing both numerator and denominator by 5, we get -4/67. So, the left side of the equation simplifies to -4/67.

Simplifying the Right Side

Now, let's simplify the right side of the equation: (-36/67)/9. Dividing by 9 is the same as multiplying by 1/9, so we have (-36/67) * (1/9) = -36/603. Simplifying this fraction by dividing both numerator and denominator by 9, we get -4/67. Thus, the right side of the equation also simplifies to -4/67.

Comparing Both Sides

We have found that both the left side and the right side of the equation simplify to -4/67. This confirms that our solution, x = -36/67, is correct. The verification process is a crucial step in problem-solving, as it ensures the accuracy of the solution and reinforces the understanding of the equation-solving process.

Conclusion

In conclusion, solving the equation (8x + 4)/5 = x/9 involves a series of algebraic steps, including cross-multiplication, distribution, isolating the variable, and verifying the solution. We found that the value of x that satisfies the equation is -36/67. This detailed walkthrough demonstrates the importance of understanding fundamental algebraic principles and techniques. By mastering these skills, you can confidently tackle a wide range of mathematical problems. The ability to solve linear equations is a cornerstone of mathematics and has practical applications in numerous fields. We hope this comprehensive guide has provided you with the clarity and confidence to solve similar equations in the future. Remember, practice is key to mastering these skills, so continue to challenge yourself with new problems and applications.