Solving The System Of Linear Equations X+y=4 And X-y=6

Solving systems of linear equations is a fundamental concept in mathematics with applications spanning various fields, including engineering, economics, and computer science. These systems arise when we have multiple equations with multiple variables, and our goal is to find the values of the variables that satisfy all equations simultaneously. In this comprehensive guide, we will delve into the process of solving a specific system of linear equations:

x + y = 4
x - y = 6

We will explore the underlying principles, the steps involved, and the rationale behind each step, ensuring a clear understanding of how to arrive at the solution. Whether you're a student grappling with linear algebra or a professional seeking a refresher, this guide will equip you with the knowledge and skills to tackle similar problems with confidence.

Understanding the Problem: Systems of Linear Equations

Before we dive into the solution, let's establish a solid foundation by understanding what a system of linear equations is and what it means to solve one. A linear equation is an equation where the highest power of any variable is 1. For instance, x + y = 4 and x - y = 6 are both linear equations. A system of linear equations is a collection of two or more linear equations involving the same variables. The solution to a system of linear equations is the set of values for the variables that make all the equations in the system true.

In our case, we have two equations with two variables, x and y. Geometrically, each linear equation in two variables represents a straight line in a two-dimensional plane. The solution to the system corresponds to the point where the two lines intersect. If the lines are parallel, there is no solution (the system is inconsistent). If the lines coincide, there are infinitely many solutions (the system is dependent).

Methods for Solving Systems of Linear Equations

There are several methods for solving systems of linear equations, each with its own advantages and disadvantages. The most common methods include:

1. Substitution: This method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily solved. After finding the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

2. Elimination (or Addition/Subtraction): This method involves manipulating the equations so that the coefficients of one of the variables are opposites. Then, by adding the equations together, we eliminate that variable, again reducing the system to a single equation with one variable. Once we solve for one variable, we can substitute it back into one of the original equations to find the other.

3. Graphing: This method involves plotting the lines represented by the equations on a coordinate plane. The point of intersection of the lines represents the solution to the system. This method is useful for visualizing the solution, but it may not be as accurate as the algebraic methods, especially when the solution involves non-integer values.

4. Matrix Methods: For larger systems of equations, matrix methods such as Gaussian elimination or matrix inversion are often used. These methods are more efficient for systems with many variables and equations.

For the given system of equations, we will use the elimination method, as it appears to be the most straightforward approach.

Solving the System Using Elimination

Let's revisit our system of equations:

x + y = 4
x - y = 6

Notice that the coefficients of y in the two equations are already opposites (+1 and -1). This makes the elimination method particularly convenient. To eliminate y, we simply add the two equations together:

(x + y) + (x - y) = 4 + 6

Simplifying the equation, we get:

2x = 10

Now, we can solve for x by dividing both sides by 2:

x = 5

We have found the value of x. Now, we need to find the value of y. We can substitute the value of x (which is 5) into either of the original equations. Let's use the first equation:

x + y = 4

Substitute x = 5:

5 + y = 4

Subtract 5 from both sides:

y = -1

Thus, we have found the value of y, which is -1.

The Solution and Its Verification

We have found that x = 5 and y = -1. Therefore, the solution to the system of equations is the ordered pair (5, -1). This means that the point (5, -1) is the intersection point of the two lines represented by the equations.

To verify our solution, we can substitute the values of x and y back into both original equations to ensure they are satisfied:

For the first equation, x + y = 4:

5 + (-1) = 4
4 = 4  (True)

For the second equation, x - y = 6:

5 - (-1) = 6
5 + 1 = 6
6 = 6  (True)

Since the solution satisfies both equations, we have verified that (5, -1) is indeed the correct solution.

Graphical Representation

To further solidify our understanding, let's visualize the solution graphically. We can rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:

x + y = 4  =>  y = -x + 4
x - y = 6  =>  y = x - 6

The first equation represents a line with a slope of -1 and a y-intercept of 4. The second equation represents a line with a slope of 1 and a y-intercept of -6. If we plot these lines on a coordinate plane, we will see that they intersect at the point (5, -1), confirming our algebraic solution.

Applications of Systems of Linear Equations

Systems of linear equations are not just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

1. Economics: Supply and demand curves can be represented as linear equations. The point where these curves intersect represents the market equilibrium, where the quantity supplied equals the quantity demanded.

2. Engineering: In circuit analysis, Kirchhoff's laws lead to systems of linear equations that can be solved to determine the currents and voltages in different parts of the circuit.

3. Computer Graphics: Linear equations are used in computer graphics for transformations such as scaling, rotation, and translation of objects.

4. Mixture Problems: Many mixture problems, such as determining the amounts of different solutions to mix to achieve a desired concentration, can be modeled using systems of linear equations.

5. Curve Fitting: Finding the equation of a line or curve that best fits a set of data points often involves solving a system of linear equations.

Conclusion: Mastering Systems of Linear Equations

In this guide, we have explored the process of solving a system of linear equations, specifically the system:

x + y = 4
x - y = 6

We have demonstrated the elimination method, which led us to the solution x = 5 and y = -1. We verified our solution by substituting the values back into the original equations and also discussed the graphical representation of the solution.

Mastering the techniques for solving systems of linear equations is crucial for success in various mathematical and scientific disciplines. By understanding the underlying principles and practicing different methods, you can develop the skills to tackle a wide range of problems involving linear relationships. Remember, the key is to break down the problem into smaller, manageable steps and to apply the appropriate method based on the specific characteristics of the system. With practice and perseverance, you can become proficient in solving systems of linear equations and unlock their power in solving real-world problems.

This foundational knowledge opens doors to more advanced topics in linear algebra and its applications. Whether you are solving for equilibrium in economic models, designing engineering systems, or working with computer graphics, the ability to solve systems of linear equations is an invaluable asset.

Continue to explore and practice, and you'll find that the world of linear equations is both fascinating and highly applicable to the world around us.