Solving For X When Y Is 11 In The Equation 7y + 109 = 9x + 12y

Introduction

In this mathematics article, we will delve into the process of solving a linear equation with two variables. Specifically, we will focus on the equation $7y + 109 = 9x + 12y$, and our objective is to determine the value of $x$ when the value of $y$ is given as 11. This type of problem is fundamental in algebra and has applications in various fields, including engineering, physics, and economics. Understanding how to solve such equations is a crucial skill for anyone pursuing studies or careers in these areas. The method we will employ involves substituting the given value of $y$ into the equation and then simplifying the equation to isolate $x$. This step-by-step approach will not only lead us to the solution but also reinforce the basic principles of algebraic manipulation. Before we dive into the solution, it's important to understand the key concepts involved, such as variables, constants, and the properties of equality. A variable is a symbol (usually a letter) that represents an unknown value, while a constant is a fixed value. The properties of equality allow us to perform the same operations on both sides of an equation without changing its balance. For example, we can add, subtract, multiply, or divide both sides of an equation by the same number, and the equation will remain true. With these concepts in mind, we are well-prepared to tackle the problem at hand.

Problem Statement

Our primary goal is to find the value of x in the given equation when y is equal to 11. The equation we are working with is:

$7y + 109 = 9x + 12y$

We are given that $y = 11$. The task at hand requires us to substitute this value of $y$ into the equation and then solve for $x$. This is a classic problem in algebra that tests our understanding of variable substitution and equation solving techniques. It is essential to approach this problem systematically, ensuring that each step is performed accurately. The first step involves replacing every instance of $y$ in the equation with the value 11. This will transform the equation from one with two variables ($x$ and $y$) to one with only one variable ($x$), making it solvable. Once we have substituted the value, we will simplify the equation by performing any necessary arithmetic operations, such as addition, subtraction, multiplication, or division. The goal is to isolate $x$ on one side of the equation, which means getting $x$ by itself. This often involves using inverse operations. For example, if $x$ is being added to a number, we would subtract that number from both sides of the equation. If $x$ is being multiplied by a number, we would divide both sides of the equation by that number. By following these steps carefully, we can determine the value of $x$ that satisfies the equation when $y = 11$. This process demonstrates the fundamental principles of algebraic problem-solving, which are crucial for more advanced mathematical concepts.

Step-by-step Solution

To determine the value of x when y equals 11, we need to follow a series of algebraic steps. These steps ensure that we accurately solve the equation while maintaining its balance. The process begins with substituting the given value of y into the original equation and progresses through simplification and isolation of the variable x.

1. Substitute y = 11 into the equation: We start by replacing every instance of y in the equation $7y + 109 = 9x + 12y$ with the value 11. This substitution is a crucial first step in solving for x, as it reduces the equation to a single variable. This gives us: $7(11) + 109 = 9x + 12(11)$

2. Simplify both sides of the equation: Next, we perform the multiplication operations on both sides of the equation. This involves multiplying 7 by 11 and 12 by 11. Performing these calculations simplifies the equation and brings us closer to isolating x. We have: $77 + 109 = 9x + 132$

   Now, we add the constants on the left side of the equation: $186 = 9x + 132$

3. Isolate the term with x: To isolate the term containing x (which is 9x), we need to eliminate the constant term (132) on the right side of the equation. We achieve this by subtracting 132 from both sides of the equation. This maintains the balance of the equation while moving us closer to solving for x. This step results in: $186 - 132 = 9x$

   Which simplifies to: $54 = 9x$

4. Solve for x: Finally, to solve for x, we need to isolate x by dividing both sides of the equation by the coefficient of x, which is 9. This division is the final step in determining the value of x. Dividing both sides by 9 gives us: $54 / 9 = x$

   Therefore: $x = 6$

Final Answer

Therefore, after substituting y = 11 into the equation $7y + 109 = 9x + 12y$ and following the steps of simplification and isolation, we have determined that the value of $x$ is 6. This solution satisfies the given equation when $y$ is equal to 11. This result is a specific solution to the equation under the given condition and demonstrates the application of algebraic principles in solving linear equations with two variables. Understanding these principles is essential for tackling more complex mathematical problems and real-world applications. The process we followed involved a systematic approach, ensuring that each step was performed accurately to arrive at the correct solution. This methodical approach is crucial in mathematics, where precision and accuracy are paramount. In conclusion, the value of x when y equals 11 in the equation $7y + 109 = 9x + 12y$ is 6. This completes the solution to the problem.