Solve 9x Y 45 For Y

Solving for 'y': A Comprehensive Guide to 9x + y = 45

This article provides a detailed explanation of how to solve the equation 9x + y = 45 for 'y', covering various approaches and highlighting key algebraic concepts. We will explore different methods, discuss their applications, and offer practice problems to solidify your understanding. This guide is designed for students of various mathematical backgrounds, from beginners needing a fundamental understanding to those seeking a more in-depth analysis.

Understanding the Equation: 9x + y = 45

Before diving into the solution, let's break down the equation itself. 9x + y = 45 is a linear equation in two variables, 'x' and 'y'. This means that when graphed, it represents a straight line. The equation states that the sum of 9 times 'x' and 'y' is equal to 45. Our goal is to isolate 'y' on one side of the equation, expressing it in terms of 'x'. This will give us a formula that allows us to calculate the value of 'y' for any given value of 'x'.

Method 1: Subtraction Property of Equality

This is the most straightforward method. The core principle here is the subtraction property of equality, which states that if you subtract the same quantity from both sides of an equation, the equation remains true.

Steps:

1. Identify the term with 'y': In our equation, this is '+y'.

2. Isolate 'y': To isolate 'y', we need to eliminate the '9x' term from the left side. We can achieve this by subtracting '9x' from both sides of the equation:

   9x + y - 9x = 45 - 9x

3. Simplify: This simplifies to:

   y = 45 - 9x

Therefore, the solution is y = 45 - 9x. This equation tells us that 'y' is dependent on 'x'. For every value of 'x', we can calculate a corresponding value of 'y'.

Method 2: Rearranging Terms

This method is essentially the same as Method 1 but emphasizes the rearrangement of terms to isolate 'y'.

Steps:

1. Rewrite the equation: Start with the original equation: 9x + y = 45

2. Move the '9x' term: To isolate 'y', we move the '9x' term to the right-hand side. Remember, when moving a term to the other side of the equation, its sign changes. Therefore, we subtract '9x' from both sides:

   y = 45 - 9x

The result is the same: y = 45 - 9x. This method highlights the flexibility in manipulating equations to achieve the desired form.

Understanding the Solution: y = 45 - 9x

The solution, y = 45 - 9x, represents a linear relationship between 'x' and 'y'. This equation can be used to find the value of 'y' for any given value of 'x'. Let's look at some examples:

• If x = 0: y = 45 - 9(0) = 45
• If x = 1: y = 45 - 9(1) = 36
• If x = 2: y = 45 - 9(2) = 27
• If x = 5: y = 45 - 9(5) = 0
• If x = 6: y = 45 - 9(6) = -9

These examples illustrate how 'y' decreases as 'x' increases. This negative relationship is reflected in the negative slope (-9) of the line represented by the equation.

Graphical Representation

The equation y = 45 - 9x can be easily graphed. The y-intercept (the point where the line crosses the y-axis) is 45, and the slope is -9. This means that for every one-unit increase in 'x', 'y' decreases by 9 units. Plotting several points (like the examples above) and connecting them will reveal a straight line sloping downwards from left to right.

Applications of Linear Equations

Linear equations like 9x + y = 45 have widespread applications in various fields:

• Physics: Modeling relationships between physical quantities like distance, velocity, and time.
• Engineering: Designing structures and systems, analyzing forces and stresses.
• Economics: Representing supply and demand curves, analyzing economic models.
• Computer Science: Developing algorithms and solving optimization problems.
• Finance: Calculating interest, analyzing investments, and forecasting financial trends.

Practice Problems

To reinforce your understanding, try solving these similar problems:

1. Solve for 'y': 3x + y = 12
2. Solve for 'y': -2x + y = 8
3. Solve for 'x': y = 2x - 7 (This problem requires you to solve for 'x' instead of 'y', a valuable exercise in reversing the process)
4. Solve for 'y': 5x - 2y = 10 (This involves a slight variation requiring an additional step before isolating 'y')

Advanced Concepts: Systems of Equations

The equation 9x + y = 45 can be part of a system of equations. A system of equations involves two or more equations with the same variables. Solving a system of equations means finding the values of the variables that satisfy all equations simultaneously. For example:

9x + y = 45 x - y = 5

To solve this system, you could use methods like substitution or elimination. Substitution involves solving one equation for one variable and substituting it into the other equation. Elimination involves adding or subtracting the equations to eliminate one variable.

Solving systems of equations is a crucial skill in various mathematical and scientific disciplines.

Conclusion

Solving the equation 9x + y = 45 for 'y' is a fundamental algebraic exercise. This article has demonstrated two primary methods—using the subtraction property of equality and rearranging terms—both leading to the solution y = 45 - 9x. Understanding this solution and its graphical representation is vital for comprehending linear equations and their broad applications across multiple fields. By practicing the provided examples and exploring advanced concepts like systems of equations, you can build a strong foundation in algebra and enhance your problem-solving abilities. Remember, consistent practice and a solid grasp of algebraic principles are key to mastering these types of problems.