Solving For Y In 9x + 5y = 9 A Step-by-Step Guide

by ADMIN 50 views

In the realm of mathematics, solving equations for specific variables is a fundamental skill. It's a cornerstone of algebra and essential for tackling more complex mathematical problems. This article delves into the process of solving the linear equation 9x + 5y = 9 for the variable y. We'll break down each step with clarity, ensuring that you not only understand the solution but also grasp the underlying principles. Whether you're a student grappling with algebra or simply someone looking to refresh your mathematical abilities, this guide offers a comprehensive approach.

Understanding Linear Equations

Before diving into the solution, let's first establish a firm understanding 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. The equation represents a straight line when graphed on a coordinate plane. The general form of a linear equation in two variables (x and y) is Ax + By = C, where A, B, and C are constants. Our equation, 9x + 5y = 9, fits this form perfectly, with A = 9, B = 5, and C = 9.

Solving for a specific variable in a linear equation means isolating that variable on one side of the equation. This involves using algebraic manipulations, such as addition, subtraction, multiplication, and division, while maintaining the equality of both sides. The goal is to express the variable you're solving for (in this case, y) in terms of the other variable (x) and constants. This process is crucial in various mathematical contexts, including finding the slope-intercept form of a line, solving systems of equations, and modeling real-world scenarios.

Step-by-Step Solution for y

Now, let's embark on the journey of solving the equation 9x + 5y = 9 for y. We'll meticulously dissect each step, ensuring a clear and concise understanding. The core principle we'll employ is the concept of inverse operations – performing the opposite operation to isolate the desired variable.

Step 1: Isolate the term with y

Our initial objective is to isolate the term containing y (which is 5y) on one side of the equation. To achieve this, we need to eliminate the term 9x from the left side. We can accomplish this by subtracting 9x from both sides of the equation. Remember, maintaining balance is paramount in equation solving; whatever operation we perform on one side, we must perform on the other.

9x + 5y - 9x = 9 - 9x

This simplifies to:

5y = 9 - 9x

We've successfully isolated the term 5y on the left side, bringing us closer to our goal of solving for y.

Step 2: Isolate y

Now that we have 5y isolated, our next task is to isolate y itself. The coefficient of y is 5, which means y is being multiplied by 5. To undo this multiplication, we perform the inverse operation: division. We'll divide both sides of the equation by 5.

(5y) / 5 = (9 - 9x) / 5

This simplifies to:

y = (9 - 9x) / 5

We've successfully isolated y! This equation expresses y in terms of x. We have effectively solved the equation for y.

Step 3: Expressing in Slope-Intercept Form (Optional)

While we've solved for y, it's often beneficial to express the equation in slope-intercept form. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope of the line and b represents the y-intercept. This form provides valuable insights into the graphical representation of the equation.

To convert our equation y = (9 - 9x) / 5 into slope-intercept form, we can simply rearrange the terms and distribute the division:

y = (9/5) - (9x/5)

y = (-9/5)x + (9/5)

Now, the equation is in slope-intercept form. We can readily identify the slope as -9/5 and the y-intercept as 9/5. This form allows us to quickly visualize the line represented by the equation.

Verification of the Solution

In mathematics, verification is a crucial step to ensure the accuracy of our solution. To verify that our solution for y is correct, we can substitute it back into the original equation and check if the equation holds true. Let's substitute y = (9 - 9x) / 5 back into the original equation 9x + 5y = 9.

9x + 5[(9 - 9x) / 5] = 9

Simplifying the equation:

9x + (9 - 9x) = 9

9x + 9 - 9x = 9

9 = 9

The equation holds true! This confirms that our solution for y is correct. Verification is an essential practice that reinforces our confidence in the solution and helps prevent errors.

Applications and Importance

Solving for variables in linear equations is not merely an academic exercise; it has numerous practical applications across various fields. Understanding how to manipulate equations to isolate variables is fundamental to:

  • Mathematics: Solving systems of equations, graphing lines, and analyzing functions all rely on this skill.
  • Science: Many scientific formulas involve multiple variables, and solving for a specific variable is often necessary to make calculations or predictions. For example, in physics, you might need to solve for velocity given distance and time.
  • Engineering: Engineers use equations extensively to design structures, circuits, and systems. Solving for specific parameters is crucial in these design processes.
  • Economics: Economic models often involve equations that relate various economic variables. Solving for a particular variable can help economists understand and predict economic trends.
  • Everyday Life: Even in everyday situations, the ability to manipulate equations can be useful. For example, if you're trying to calculate how much time you need to save to reach a financial goal, you might need to solve for time in a savings equation.

The ability to solve for variables is a powerful tool that empowers you to tackle a wide range of problems in mathematics and beyond. It's a skill that builds confidence and enhances your problem-solving capabilities.

Common Mistakes to Avoid

While the process of solving for y in a linear equation is relatively straightforward, certain common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and ensure accuracy in your work.

  1. Incorrectly Applying Inverse Operations: A frequent error is not applying the correct inverse operation. For instance, instead of dividing both sides by 5 to undo multiplication, students might subtract 5. Always remember to perform the opposite operation to isolate the variable.
  2. Not Maintaining Balance: A cardinal rule of equation solving is maintaining balance. Any operation performed on one side of the equation must be performed on the other side. Failing to do so will disrupt the equality and lead to an incorrect solution.
  3. Sign Errors: Sign errors are common culprits in algebraic mistakes. Pay close attention to the signs of the terms when adding, subtracting, multiplying, or dividing. A misplaced negative sign can throw off the entire solution.
  4. Incorrectly Distributing: When dealing with expressions in parentheses, remember to distribute correctly. Ensure that you multiply each term inside the parentheses by the factor outside.
  5. Forgetting to Verify: As emphasized earlier, verification is a critical step. Neglecting to substitute your solution back into the original equation can leave errors undetected.

By being mindful of these common mistakes and diligently applying the principles of equation solving, you can significantly reduce the likelihood of errors and achieve accurate results.

Conclusion

In this comprehensive guide, we've meticulously walked through the process of solving the equation 9x + 5y = 9 for y. We began by establishing a solid understanding of linear equations and the importance of solving for variables. We then systematically dissected the steps involved in isolating y, emphasizing the use of inverse operations and maintaining balance in the equation. Furthermore, we explored the benefits of expressing the solution in slope-intercept form and underscored the significance of verifying the solution.

We also delved into the wide-ranging applications of solving for variables, highlighting its relevance in mathematics, science, engineering, economics, and even everyday life. By recognizing the practical value of this skill, you can appreciate its importance and cultivate a stronger motivation to master it.

Finally, we addressed common mistakes that students often encounter when solving equations. By being aware of these pitfalls, you can proactively avoid them and enhance your accuracy. Solving for variables is a fundamental skill that forms the bedrock of algebraic proficiency. With practice and a clear understanding of the principles involved, you can confidently tackle a wide array of mathematical challenges.

This guide serves as a valuable resource for students, educators, and anyone seeking to strengthen their algebraic skills. Embrace the power of equation solving, and unlock a world of mathematical possibilities.