Solving Linear Equations Value Of X In System

by Admin 46 views

In the realm of mathematics, solving systems of linear equations is a fundamental skill. These systems appear in various applications, from modeling real-world phenomena to solving complex engineering problems. In this article, we will delve into the process of solving a system of linear equations using the elimination method. We will follow Veda's approach to solve the given system and determine the value of x in the solution.

Understanding the System of Equations

The system of equations we aim to solve is:

6 + 4x - 2y = 0
-3 - 7y = 10x

These are two linear equations with two variables, x and y. To find the solution, we need to find the values of x and y that satisfy both equations simultaneously. The elimination method is a powerful technique for solving such systems.

The Elimination Method: A Step-by-Step Approach

The elimination method involves manipulating the equations in the system to eliminate one of the variables. This is achieved by multiplying one or both equations by a constant so that the coefficients of one variable are opposites. When the equations are added together, that variable is eliminated, leaving us with a single equation in one variable.

Step 1: Rearrange the Equations

Before applying the elimination method, it's often helpful to rearrange the equations into a standard form, where the variables are on one side and the constants are on the other. Let's rewrite the given equations:

Equation 1: 4x - 2y = -6 Equation 2: 10x + 7y = -3

Step 2: Choose a Variable to Eliminate

We can choose to eliminate either x or y. Let's choose to eliminate x. To do this, we need to find a common multiple of the coefficients of x (4 and 10). The least common multiple of 4 and 10 is 20.

Step 3: Multiply the Equations

To make the coefficients of x opposites, we'll multiply Equation 1 by -5 and Equation 2 by 2:

Equation 1 × (-5): -20x + 10y = 30 Equation 2 × (2): 20x + 14y = -6

Step 4: Add the Equations

Now, we add the modified equations together. Notice that the x terms cancel out:

(-20x + 10y) + (20x + 14y) = 30 + (-6) 24y = 24

Step 5: Solve for the Remaining Variable

We now have a simple equation in one variable. Solving for y:

y = 24 / 24 y = 1

Step 6: Substitute to Find the Other Variable

Now that we know the value of y, we can substitute it back into either of the original equations to solve for x. Let's use Equation 1:

4x - 2(1) = -6 4x - 2 = -6 4x = -4 x = -1

The Solution

Therefore, the solution to the system of equations is x = -1 and y = 1. The question specifically asks for the value of x, which is -1. This methodical approach to solving linear equations showcases the power and precision of algebraic techniques. By carefully manipulating equations, we can systematically isolate variables and arrive at the solution. This skill is not only crucial in mathematics but also finds applications in various scientific and engineering disciplines.

Importance of Linear Equations in Mathematics

Linear equations form the bedrock of numerous mathematical concepts and real-world applications. Understanding how to solve them is paramount for anyone delving into mathematics, science, engineering, or economics. The elimination method, as demonstrated in solving this system, is one of the most versatile and widely used techniques. Its strength lies in its systematic approach, allowing us to tackle even complex systems with relative ease.

Applications of Linear Equations

Linear equations are not just abstract mathematical constructs; they are powerful tools for modeling and solving real-world problems. Here are a few examples:

  • Modeling Physical Systems: Linear equations can describe the relationships between physical quantities like distance, time, and velocity, or force, mass, and acceleration. Engineers use them to design structures, analyze circuits, and model fluid flow.
  • Economic Analysis: Economists use systems of linear equations to model supply and demand, analyze market equilibrium, and predict economic trends. Linear programming, a technique for optimizing linear functions subject to linear constraints, is a cornerstone of operations research and management science.
  • Computer Graphics: Linear transformations, which are represented by linear equations, are the foundation of computer graphics. They are used to rotate, scale, and translate objects in 3D space, creating the visual effects we see in video games and movies.
  • Data Analysis: Linear regression, a statistical technique based on linear equations, is used to find the best-fitting line through a set of data points. This allows us to identify trends, make predictions, and understand relationships between variables.

Advantages of the Elimination Method

The elimination method offers several advantages over other methods for solving systems of linear equations, such as substitution or graphing:

  • Systematic Approach: The method follows a clear and logical sequence of steps, making it less prone to errors.
  • Versatility: It can be applied to systems with any number of equations and variables, as long as the system is linear.
  • Efficiency: For larger systems, elimination is often more efficient than substitution.
  • Foundation for Advanced Techniques: The principles of elimination underlie more advanced techniques like Gaussian elimination and matrix inversion, which are used to solve even more complex systems.

Common Mistakes to Avoid

While the elimination method is straightforward, there are some common mistakes to watch out for:

  • Incorrect Multiplication: Make sure to multiply every term in the equation by the constant, not just the terms with the variable you're trying to eliminate.
  • Sign Errors: Be careful with signs when adding or subtracting equations. A simple sign error can throw off the entire solution.
  • Arithmetic Mistakes: Double-check your arithmetic calculations, especially when dealing with fractions or decimals.
  • Forgetting to Substitute: After solving for one variable, remember to substitute the value back into one of the original equations to find the other variable.

By understanding the principles behind the elimination method and avoiding these common pitfalls, you can confidently solve a wide range of systems of linear equations.

Conclusion

In summary, we have successfully used the elimination method to solve the given system of linear equations. We found that the value of x in the solution is -1. This exercise highlights the importance of understanding and applying algebraic techniques to solve mathematical problems. Linear equations are fundamental tools with wide-ranging applications, and mastering methods like elimination is crucial for success in mathematics and related fields. The value of x, which we meticulously calculated, stands as a testament to the power of systematic problem-solving in mathematics. By understanding the underlying principles and applying them with precision, we can unlock solutions to complex problems and gain a deeper appreciation for the beauty and utility of mathematics.

This detailed walkthrough not only provides the solution but also emphasizes the broader context and significance of linear equations and the elimination method in mathematics and beyond. The step-by-step approach ensures clarity and understanding, while the discussion of applications and common mistakes enhances the learning experience.