Solving Linear Equations Find The Value Of Y
We are given the following system of linear equations:
Our goal is to find the value of $y$ when $x = 3$. This involves substituting the given value of $x$ into one of the equations and solving for $y$. This problem highlights a fundamental concept in algebra: solving systems of linear equations. These equations represent lines, and the solution to the system represents the point where the lines intersect. Understanding how to solve such systems is crucial for various applications in mathematics, science, and engineering. For example, linear equations can model relationships between quantities, predict outcomes, and optimize processes. In this specific case, we will demonstrate how substitution allows us to isolate a variable and find its value. Solving systems of equations is not just a mathematical exercise; it's a tool for understanding and modeling the world around us. From balancing chemical equations to predicting economic trends, the principles of linear algebra are essential. By mastering these techniques, we can unlock the power of mathematics to solve complex problems.
Substituting the Value of x
To find the value of $y$, we will substitute $x = 3$ into the first equation:
This simplifies to:
The next step is to isolate $y$ by subtracting 6 from both sides of the equation. This is a standard algebraic technique used to solve for unknowns. The principle behind it is maintaining the balance of the equation: whatever operation is performed on one side must also be performed on the other. This ensures that the equality remains valid throughout the process. In this case, subtracting 6 from both sides allows us to eliminate the constant term on the left side, bringing us closer to isolating $y$. This process of isolating a variable is fundamental to solving equations in algebra and beyond. It's a powerful tool that allows us to determine the value of an unknown quantity based on its relationship with other known quantities. By mastering this technique, we can tackle a wide range of mathematical problems and real-world applications. So, let's proceed with subtracting 6 from both sides to continue our journey towards finding the value of $y$. Remember, each step we take is a logical progression, building upon the previous one, to arrive at the solution.
Solving for y
Subtracting 6 from both sides, we get:
Therefore, the value of $y$ when $x = 3$ is 3. This result confirms that the point (3, 3) lies on the line represented by the equation 2x + y = 9. Furthermore, if we were to substitute these values into the second equation, 2x - y = 3, we would find that 2(3) - 3 = 3, which is also true. This demonstrates that (3, 3) is indeed the solution to the system of equations, representing the point where the two lines intersect. Solving for $y$ involves isolating the variable, which is a crucial skill in algebra and beyond. It's a fundamental technique used in various mathematical and scientific applications. The ability to manipulate equations and isolate variables allows us to solve for unknowns and gain insights into the relationships between different quantities. In this case, we have successfully found the value of $y$ by applying basic algebraic principles. This reinforces the importance of understanding these principles for problem-solving and analytical thinking.
The value of $y$ is 3.
Answer Options
Given the options:
A. -2 B. 2 C. 3 D. -3
The correct answer is:
C. 3
Understanding the Solution
We found the value of $y$ by using a straightforward substitution method. Substituting known values into equations is a fundamental skill in algebra. This method is widely applicable in various mathematical problems and real-world scenarios. When dealing with systems of equations, substitution allows us to reduce the number of variables and solve for the unknowns one by one. It's a powerful technique that can be used to solve for any variable in the system, as long as we have enough information. The key is to carefully substitute the values and simplify the equations step by step. By mastering substitution, we can tackle more complex problems involving multiple equations and variables. This technique is not only essential in mathematics but also in fields such as physics, engineering, and economics, where mathematical models are used to represent real-world phenomena. Therefore, understanding and applying substitution is a crucial skill for problem-solving and analytical thinking.
Importance of Linear Equations
Linear equations are the backbone of many mathematical models. They are used to represent relationships between variables in a simple and understandable way. The ability to solve linear equations is essential for understanding more complex mathematical concepts. For example, in calculus, linear approximations are used to estimate the values of functions. In linear algebra, systems of linear equations are used to solve for unknowns in matrices and vectors. The applications of linear equations are vast and varied, ranging from simple arithmetic problems to complex scientific simulations. Understanding the properties and solutions of linear equations is crucial for anyone pursuing a career in mathematics, science, or engineering. It's a foundational concept that builds the basis for further mathematical exploration and problem-solving. Therefore, mastering linear equations is not just about solving problems; it's about building a strong foundation for mathematical reasoning and analytical thinking.
Solving for Variables
Solving for variables is a core skill in algebra. It involves isolating the variable of interest by performing algebraic operations on both sides of the equation. This process requires a clear understanding of the properties of equality and the order of operations. For example, to solve for $y$ in the equation $2x + y = 9$, we first substitute the value of $x$, then subtract the constant term from both sides. The goal is to get the variable alone on one side of the equation, so its value can be easily determined. Solving for variables is not just a mathematical exercise; it's a fundamental skill for problem-solving in various contexts. It's used in science to solve for physical quantities, in economics to solve for market equilibrium, and in engineering to design structures and systems. The ability to manipulate equations and isolate variables is a crucial skill for anyone who wants to apply mathematical concepts to real-world problems. Therefore, mastering this skill is essential for success in many fields.
Practice and Application
To master the skill of solving linear equations, consistent practice is essential. Working through various problems helps to reinforce the concepts and develop problem-solving strategies. It's also important to apply these skills to real-world scenarios to understand their practical relevance. For example, linear equations can be used to model the cost of goods, the distance traveled at a constant speed, or the relationship between supply and demand. By applying these concepts to real-world situations, we can gain a deeper understanding of their meaning and significance. Practice and application are key to building confidence and competence in solving linear equations and other mathematical problems. It's a process of continuous learning and refinement, where each problem solved builds upon the previous one. Therefore, consistent effort and a willingness to apply these skills are crucial for success.
In summary, solving for $y$ in the given linear equation system when $x = 3$ involves substituting the value of $x$ into one of the equations and solving for $y$. This process demonstrates the fundamental principles of algebra and the importance of mastering these skills for various applications.
