Solving For Y In The Equation 6x-3y=9: A Step By Step Guide

In the realm of mathematics, a fundamental skill lies in solving equations for a specific variable. This article provides a comprehensive guide on how to solve for 'y' in the linear equation 6x - 3y = 9. We will explore the step-by-step process of isolating 'y', ensuring a clear understanding of the algebraic manipulations involved. This process is crucial for various mathematical applications, including graphing linear equations, solving systems of equations, and understanding the relationships between variables. This article aims to equip you with the knowledge and confidence to tackle similar problems effectively. By mastering the technique of isolating variables, you'll unlock a deeper understanding of mathematical relationships and gain a powerful tool for problem-solving across diverse contexts. The ability to manipulate equations and express variables in terms of others is a cornerstone of algebraic thinking and essential for success in more advanced mathematical studies. So, let's embark on this journey of algebraic exploration and uncover the secrets of solving for 'y'.

Understanding the Equation: 6x - 3y = 9

The equation 6x - 3y = 9 is a linear equation in two variables, 'x' and 'y'. Linear equations represent a straight line when graphed on a coordinate plane. The goal of solving for 'y' is to rewrite the equation in the slope-intercept form, which is y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept. This form allows us to easily visualize and analyze the behavior of the line. The coefficients of 'x' and 'y' play a significant role in determining the line's slope and orientation. The constant term, in this case, 9, influences the line's position on the coordinate plane. To isolate 'y', we must carefully perform algebraic operations on both sides of the equation, ensuring that the equality remains balanced. This involves strategically moving terms and coefficients until 'y' stands alone on one side of the equation. The process of solving for 'y' not only reveals its relationship with 'x' but also provides insights into the equation's graphical representation. By understanding the slope-intercept form, we can quickly sketch the line and interpret its characteristics. Furthermore, solving for 'y' is a critical step in solving systems of linear equations, where we seek the points of intersection between two or more lines. The ability to manipulate linear equations and isolate variables is a fundamental skill in algebra and serves as a building block for more advanced mathematical concepts.

Step-by-Step Solution: Isolating 'y'

To isolate 'y' in the equation 6x - 3y = 9, we'll follow these steps:

Step 1: Subtract 6x from both sides.

Our goal is to isolate the term containing 'y' on one side of the equation. To achieve this, we need to eliminate the 6x term from the left side. We can accomplish this by subtracting 6x from both sides of the equation. This maintains the balance of the equation, as we are performing the same operation on both sides. The equation now becomes: 6x - 3y - 6x = 9 - 6x. Simplifying this, we get -3y = 9 - 6x. This step effectively moves the 6x term to the right side, leaving the term with 'y' isolated on the left. The importance of performing the same operation on both sides cannot be overemphasized, as it is the cornerstone of maintaining equality in algebraic manipulations. By subtracting 6x, we have taken a significant stride towards isolating 'y' and bringing the equation closer to the desired form. This step demonstrates the fundamental principle of using inverse operations to isolate variables in equations.

Step 2: Divide both sides by -3.

Now we have the equation -3y = 9 - 6x. To completely isolate 'y', we need to eliminate the coefficient -3 that is multiplying it. We can do this by dividing both sides of the equation by -3. This operation cancels out the -3 on the left side, leaving 'y' by itself. Dividing both sides by -3 gives us: (-3y) / -3 = (9 - 6x) / -3. Simplifying the left side, we get y. On the right side, we need to divide both terms by -3: 9 / -3 = -3 and -6x / -3 = 2x. Therefore, the equation becomes y = -3 + 2x. This step is crucial as it completes the isolation of 'y', expressing it in terms of 'x'. Dividing by the coefficient ensures that 'y' stands alone, revealing its direct relationship with 'x'. The negative sign in the coefficient plays a significant role, as dividing by a negative number changes the signs of the terms on the right side. This step highlights the importance of careful attention to signs and arithmetic operations when manipulating equations. By dividing both sides by -3, we have successfully solved for 'y', bringing us closer to understanding the equation's characteristics and behavior.

Step 3: Simplify the equation.

We now have y = -3 + 2x. While this equation is technically correct, it is conventional to write it in the slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. To achieve this, we simply rearrange the terms on the right side. Swapping the positions of -3 and 2x gives us y = 2x - 3. This is the simplified form of the equation, where 'y' is expressed explicitly in terms of 'x'. The coefficient of 'x', which is 2, represents the slope of the line, and the constant term, -3, represents the y-intercept. Writing the equation in slope-intercept form makes it easier to graph the line and understand its properties. The slope tells us the steepness and direction of the line, while the y-intercept indicates the point where the line crosses the y-axis. Simplifying the equation not only presents it in a more standard form but also reveals valuable information about the line it represents. This step emphasizes the importance of expressing equations in a clear and concise manner, facilitating further analysis and interpretation. By rearranging the terms, we have arrived at the final solution, y = 2x - 3, which is the equation solved for 'y' and written in slope-intercept form.

The Solution: y = 2x - 3

Therefore, the solution to the equation 6x - 3y = 9 when solved for 'y' is y = 2x - 3. This equation represents a line with a slope of 2 and a y-intercept of -3. Understanding this solution allows us to analyze the relationship between 'x' and 'y' and visualize the line on a coordinate plane. For every increase of 1 in 'x', 'y' increases by 2. The line crosses the y-axis at the point (0, -3). This form of the equation, y = mx + b, provides a powerful tool for understanding and manipulating linear relationships. It allows us to quickly determine the slope and y-intercept, which are crucial characteristics of a line. The solution y = 2x - 3 also enables us to find the value of 'y' for any given value of 'x', and vice versa. This is essential for solving problems involving linear equations and their applications. Moreover, this solution can be used in conjunction with other linear equations to solve systems of equations, finding the points of intersection between lines. The ability to solve for 'y' and express the equation in slope-intercept form is a fundamental skill in algebra, providing a foundation for more advanced mathematical concepts and problem-solving techniques. This solution, y = 2x - 3, encapsulates the relationship between 'x' and 'y' in a clear and concise manner, facilitating further analysis and application.

Applications and Further Practice

Solving for 'y' is a fundamental skill with numerous applications in mathematics and related fields. It is essential for graphing linear equations, solving systems of equations, and understanding the relationships between variables in various contexts. In graphing, the slope-intercept form (y = mx + b) makes it easy to plot the line by identifying the y-intercept and using the slope to find other points. When solving systems of equations, isolating 'y' in one or both equations allows for the use of substitution or elimination methods to find the solution. Moreover, this skill is crucial in various real-world applications, such as modeling linear relationships in physics, economics, and engineering. For instance, it can be used to determine the cost of a service based on a fixed fee and an hourly rate, or to analyze the relationship between supply and demand in economics. To further solidify your understanding, practice solving for 'y' in various linear equations with different coefficients and constants. Try rearranging equations with parentheses or fractions to challenge yourself. You can also explore online resources and textbooks for additional practice problems and examples. The more you practice, the more confident and proficient you will become in solving for 'y' and applying this skill to diverse mathematical problems. Mastering this skill will not only enhance your algebraic abilities but also provide a solid foundation for tackling more advanced mathematical concepts and real-world applications.

By mastering the steps outlined in this guide, you can confidently solve for 'y' in linear equations and unlock a deeper understanding of mathematical relationships.