Finding The X-Coordinate At The Intersection Of Linear Equations

Jun 24, 2025 by ADMIN 65 views

Given the system of linear equations 3x + 2y = 4 and 3x - 2y = 7, what is the value of the x-coordinate at the point of intersection?

Solving a System of Linear Equations to Find the X-Coordinate of the Intersection Point

Introduction

In mathematics, systems of linear equations are fundamental concepts, often encountered in various fields, from basic algebra to advanced engineering and economics. These systems represent relationships between variables, and their solutions provide valuable insights into the interactions between these variables. One common task is to find the point of intersection of the lines represented by these equations, which corresponds to the solution of the system. In this article, we will delve into the process of solving a system of linear equations to determine the x-coordinate of the intersection point. We will use a specific example to illustrate the steps involved, providing a clear and comprehensive guide for readers to understand and apply these techniques.

Consider the following system of linear equations:

3x + 2y = 4
3x - 2y = 7

Our goal is to find the value of the x-coordinate for the point where these two lines intersect. This point represents the solution to the system of equations, where both equations are simultaneously satisfied. To achieve this, we will employ the method of elimination, a powerful technique for solving systems of linear equations. By carefully manipulating the equations, we can eliminate one variable, allowing us to solve for the other. This process will ultimately lead us to the x-coordinate we seek.

Understanding Systems of Linear Equations

Before diving into the solution, it's crucial to understand the basics of systems 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. In a two-variable system, such as the one we are considering, the equations represent straight lines on a coordinate plane. The point of intersection of these lines, if it exists, represents the unique solution to the system. This point's coordinates satisfy both equations simultaneously.

There are several methods to solve systems of linear equations, including substitution, elimination, and graphical methods. The choice of method often depends on the specific structure of the equations. In this case, the elimination method is particularly well-suited due to the presence of terms with opposite signs for the y-variable. This allows us to easily eliminate y by adding the equations together.

Step-by-Step Solution Using Elimination Method

1. Identifying the Equations

We begin with the given system of linear equations:

3x + 2y = 4   (Equation 1)
3x - 2y = 7   (Equation 2)

2. Eliminating the Variable

Notice that the coefficients of the y-terms in the two equations are opposites (+2 and -2). This allows us to eliminate the y-variable by simply adding the two equations together. This is a key step in the elimination method, as it reduces the system to a single equation with one variable.

Adding Equation 1 and Equation 2:

(3x + 2y) + (3x - 2y) = 4 + 7

Simplifying the equation:

6x = 11

The y-terms cancel out, leaving us with a simple equation in terms of x.

3. Solving for x

To find the value of x, we divide both sides of the equation by 6:

x = 11 / 6

This gives us the exact value of x as a fraction. To express it as a decimal, we perform the division:

x ≈ 1.8333

We have now found the x-coordinate of the intersection point.

4. Finding the y-coordinate (Optional)

While the problem specifically asks for the x-coordinate, we can also find the y-coordinate to fully determine the point of intersection. To do this, we substitute the value of x back into either Equation 1 or Equation 2. Let's use Equation 1:

3(1.8333) + 2y = 4

5.5 + 2y = 4

Subtract 5.5 from both sides:

2y = -1.5

Divide by 2:

y = -0.75

So, the y-coordinate of the intersection point is approximately -0.75.

5. The Point of Intersection

The point of intersection is approximately (1.8333, -0.75). However, the question specifically asks for the x-coordinate, which we have already found.

Alternative Methods for Solving Linear Equations

While we used the elimination method in this example, it's worth noting that other methods can also be used to solve systems of linear equations. Two common alternatives are:

Substitution Method: In this method, we solve one equation for one variable and substitute that expression into the other equation. This results in a single equation with one variable, which can then be solved. The value of the solved variable is then substituted back into one of the original equations to find the value of the other variable.
Graphical Method: This method involves plotting both equations on a coordinate plane. The point where the lines intersect represents the solution to the system. This method is particularly useful for visualizing the solution and understanding the relationship between the equations.

Importance of Solving Systems of Linear Equations

Solving systems of linear equations is a fundamental skill in mathematics with wide-ranging applications. These systems appear in various real-world scenarios, including:

Engineering: In structural analysis, systems of equations are used to determine the forces and stresses in a structure. In electrical engineering, they are used to analyze circuits.
Economics: Systems of equations are used to model supply and demand, market equilibrium, and other economic phenomena.
Computer Science: Linear equations are used in computer graphics, optimization problems, and machine learning.
Everyday Life: Systems of equations can be used to solve problems involving mixtures, rates, and other practical situations.

Practical Applications and Examples

To further illustrate the importance of solving linear equations, let's consider a few practical examples:

Mixture Problems: Suppose you have two solutions with different concentrations of a substance. You want to mix them to obtain a solution with a specific concentration. This problem can be modeled using a system of linear equations.
Rate Problems: Imagine two cars traveling towards each other at different speeds. You want to find the time it takes for them to meet. This can be solved by setting up a system of equations based on the distances traveled and the speeds of the cars.
Investment Problems: Consider an investor who wants to allocate their funds between two different investment options with varying rates of return. The investor's goal is to achieve a specific overall return. This can be modeled using linear equations.

Common Mistakes and How to Avoid Them

When solving systems of linear equations, it's essential to be careful and avoid common mistakes. Some of the most frequent errors include:

Arithmetic Errors: Simple arithmetic mistakes, such as incorrect addition or subtraction, can lead to wrong solutions. Always double-check your calculations.
Sign Errors: Pay close attention to the signs of the coefficients and constants. A sign error can completely change the solution.
Incorrect Substitution: When using the substitution method, ensure that you substitute the expression correctly into the other equation.
Misinterpreting the Question: Always read the question carefully and make sure you are answering what is being asked. For example, in this problem, we only needed to find the x-coordinate, not the entire point of intersection.

To avoid these mistakes, it's helpful to:

Show Your Work: Write down each step of your solution clearly. This makes it easier to identify and correct errors.
Check Your Solution: After finding a solution, substitute the values back into the original equations to verify that they are satisfied.
Practice Regularly: The more you practice solving systems of linear equations, the more comfortable and confident you will become.

Conclusion

In conclusion, we have successfully found the x-coordinate of the intersection point for the given system of linear equations using the elimination method. The x-coordinate is approximately 1.8333. This process demonstrates the power and versatility of linear equations in solving mathematical problems. Understanding systems of linear equations and the techniques for solving them is crucial for success in various fields of study and real-world applications. By following the steps outlined in this article and practicing regularly, readers can develop a strong foundation in this important mathematical concept. Remember to always double-check your work and be mindful of common mistakes to ensure accurate solutions.

The key takeaway from this discussion is the importance of mastering the techniques for solving systems of linear equations. These skills are not only valuable in academic settings but also in numerous practical situations. By understanding the underlying principles and practicing the methods, you can confidently tackle a wide range of problems involving linear relationships. Whether you are an engineer designing a structure, an economist analyzing market trends, or simply solving a puzzle, the ability to solve linear equations will prove to be an invaluable asset.

This comprehensive guide has provided a detailed explanation of the elimination method, along with alternative approaches and practical examples. By studying this material and engaging in further practice, you can enhance your understanding of systems of linear equations and their applications. So, continue to explore the world of mathematics and discover the power of linear equations in solving complex problems.