Solving For X In 2x + 3y = 36 When Y = 6

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In the realm of mathematics, solving equations is a fundamental skill. This article will delve into the process of finding the value of x in the equation 2x + 3y = 36, given that y = 6. This type of problem falls under the category of linear equations, which are equations where the highest power of the variable is 1. Mastering these equations is crucial for various mathematical concepts and real-world applications.

Understanding Linear Equations

Before diving into the solution, let's 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. The graph of a linear equation is a straight line. The general form of a linear equation in two variables (x and y) is ax + by = c, where a, b, and c are constants. In our case, the equation 2x + 3y = 36 fits this form, with a = 2, b = 3, and c = 36. The key to solving linear equations is to isolate the variable we are trying to find (in this case, x) on one side of the equation. This involves using algebraic operations such as addition, subtraction, multiplication, and division while maintaining the equality of both sides.

One of the crucial aspects of understanding linear equations is recognizing their graphical representation. When plotted on a coordinate plane, a linear equation forms a straight line. Each point on this line represents a solution to the equation, meaning that the x and y coordinates of that point satisfy the equation. The slope of the line indicates the rate of change between x and y, while the y-intercept shows where the line crosses the y-axis. This visual representation can be incredibly helpful in understanding the relationship between variables and in solving systems of linear equations. Furthermore, linear equations are not just theoretical constructs; they have practical applications in various fields. They can be used to model relationships between quantities, such as distance and time, cost and quantity, or supply and demand. By understanding linear equations, we gain a powerful tool for analyzing and solving real-world problems. In essence, linear equations serve as a cornerstone of mathematical literacy, providing a foundation for more advanced mathematical concepts and practical problem-solving skills.

Step-by-Step Solution

Now, let's solve the equation 2x + 3y = 36 when y = 6. This involves a series of algebraic steps to isolate x and determine its value. Here's a detailed breakdown:

  1. Substitute the value of y: The first step is to replace y in the equation with its given value, which is 6. This gives us: 2x + 3(6) = 36

    This substitution transforms the equation from one with two variables (x and y) into one with a single variable (x), making it solvable.

  2. Simplify the equation: Next, we simplify the equation by performing the multiplication: 2x + 18 = 36

    This step reduces the equation to a more manageable form, making it easier to isolate x.

  3. Isolate the term with x: To isolate the term with x, we need to get rid of the constant term (+18) on the left side of the equation. We do this by subtracting 18 from both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must perform on the other side to maintain the balance.

    2x + 18 - 18 = 36 - 18 2x = 18

    Subtracting 18 from both sides ensures that the equation remains balanced and isolates the term containing x on one side.

  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 2:

    (2x) / 2 = 18 / 2 x = 9

    This division isolates x on the left side, giving us the value of x that satisfies the equation when y = 6. Thus, the solution to the equation 2x + 3y = 36 when y = 6 is x = 9. This step-by-step approach highlights the importance of maintaining balance in an equation while manipulating it to solve for an unknown variable. Each operation performed brings us closer to isolating the variable, ultimately revealing its value. The clarity of this method makes it easy to follow and apply to similar algebraic problems.

Verification

To ensure our solution is correct, we can substitute the values of x and y back into the original equation. This process, known as verification, is a crucial step in problem-solving as it confirms the accuracy of our answer. Substituting x = 9 and y = 6 into the equation 2x + 3y = 36, we get:

2(9) + 3(6) = 36 18 + 18 = 36 36 = 36

Since the equation holds true, our solution x = 9 is indeed correct. Verification is not just a formality; it's an essential practice that can prevent errors and build confidence in our problem-solving abilities. By substituting the calculated values back into the original equation, we confirm that the equation remains balanced, indicating that the solution is accurate. This step is particularly useful in more complex problems where mistakes are more likely to occur. Moreover, the habit of verification reinforces the understanding of the equation's structure and the relationship between variables. It ensures that the solution not only satisfies the algebraic manipulations but also makes logical sense within the context of the problem. Therefore, incorporating verification as a standard part of the problem-solving process is a valuable strategy for achieving accuracy and fostering a deeper understanding of mathematical concepts.

Importance of Algebraic Manipulation

This problem highlights the importance of algebraic manipulation in solving equations. Algebraic manipulation involves rearranging equations using valid operations (addition, subtraction, multiplication, division) to isolate the variable of interest. These skills are essential not only in mathematics but also in various fields like physics, engineering, and computer science.

The ability to manipulate algebraic expressions and equations is a cornerstone of mathematical proficiency. It empowers us to transform complex problems into simpler, more manageable forms. The process of algebraic manipulation involves applying a set of rules and principles to rearrange equations, simplify expressions, and isolate variables. These rules are based on fundamental mathematical properties, such as the commutative, associative, and distributive laws, as well as the properties of equality. By mastering algebraic manipulation, we gain the ability to solve a wide range of problems, from simple linear equations to more intricate systems of equations and inequalities. This skill is not limited to the realm of pure mathematics; it has practical applications in numerous fields. In physics, for example, algebraic manipulation is crucial for solving equations related to motion, forces, and energy. In engineering, it is used in circuit analysis, structural design, and control systems. In computer science, it is applied in algorithm design, data analysis, and software development. The versatility of algebraic manipulation makes it an indispensable tool for anyone pursuing a career in science, technology, engineering, or mathematics (STEM). Furthermore, the ability to manipulate algebraic expressions fosters critical thinking and problem-solving skills that are valuable in any field. It encourages a systematic approach to problem-solving, where complex problems are broken down into smaller, more manageable steps. This approach not only leads to solutions but also enhances understanding of the underlying concepts. Therefore, investing time and effort in mastering algebraic manipulation is an investment in one's future success, both academically and professionally.

Real-World Applications

Linear equations like the one we solved have numerous real-world applications. For instance, they can be used to model relationships between quantities, such as the cost of a service based on the number of hours worked, or the distance traveled by a car based on its speed and time. Understanding how to solve these equations allows us to make predictions and solve practical problems.

The applicability of linear equations extends far beyond the classroom, permeating various aspects of our daily lives and professional endeavors. These equations serve as powerful tools for modeling relationships between two or more variables, allowing us to analyze, predict, and solve real-world problems. One common application is in financial planning, where linear equations can be used to calculate budgets, track expenses, and estimate loan payments. For example, a linear equation can represent the total cost of a service, such as plumbing or tutoring, based on an hourly rate. Similarly, businesses use linear equations to forecast sales, analyze market trends, and determine pricing strategies. In the field of physics, linear equations are fundamental for describing motion, forces, and energy. They are used to calculate the distance traveled by an object moving at a constant speed, the force required to accelerate a mass, and the energy stored in a system. Engineers rely on linear equations for designing structures, analyzing circuits, and optimizing processes. For instance, they can be used to calculate the stress on a beam, the current flowing through a resistor, or the flow rate in a pipe. In computer science, linear equations play a crucial role in algorithm design, data analysis, and machine learning. They are used to develop models for predicting customer behavior, optimizing resource allocation, and classifying data points. The widespread use of linear equations highlights their importance in various disciplines and underscores the value of mastering the skills required to solve them. By understanding the principles of linear equations and their applications, we can make informed decisions, solve practical problems, and gain a deeper appreciation for the mathematical foundations of our world.

Conclusion

In conclusion, we have successfully solved for x in the equation 2x + 3y = 36 when y = 6, finding that x = 9. This exercise demonstrates the importance of understanding linear equations and algebraic manipulation. These skills are fundamental for success in mathematics and various other fields. By mastering these concepts, we can confidently tackle more complex problems and apply our knowledge to real-world situations. The ability to solve linear equations is a valuable asset that empowers us to make informed decisions, analyze data, and solve practical challenges in a wide range of contexts.