2x 3y 7 5x Y 9

Decoding the Enigma: A Deep Dive into the Mathematical Relationship of 2x + 3y = 7 and 5x + y = 9

This article explores the mathematical relationship between the two linear equations, 2x + 3y = 7 and 5x + y = 9. We will delve into various methods of solving this system of equations, analyze the significance of the solution, and discuss the broader implications within the realm of linear algebra and its applications. This comprehensive guide will appeal to students grappling with simultaneous equations, as well as anyone interested in a deeper understanding of mathematical problem-solving.

Understanding Systems of Linear Equations

Before we embark on solving our specific system, let's establish a foundational understanding of what constitutes a system 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 raised to the power of 1. In simpler terms, it represents a straight line when graphed. A system of linear equations involves two or more linear equations with the same variables. The solution to the system is the set of values for the variables that simultaneously satisfy all equations in the system. Geometrically, this represents the point(s) of intersection between the lines (in two dimensions) or planes (in three dimensions).

Our system, composed of 2x + 3y = 7 and 5x + y = 9, is a system of two linear equations with two variables (x and y). Solving this system means finding the values of x and y that make both equations true.

Methods for Solving the System of Equations

Several methods can effectively solve systems of linear equations. We will explore three prominent approaches: substitution, elimination, and graphing.

1. The Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. Let's apply this to our system:

1. Solve for one variable: From the equation 5x + y = 9, we can easily solve for y: y = 9 - 5x.

2. Substitute: Now, substitute this expression for y (9 - 5x) into the first equation (2x + 3y = 7): 2x + 3(9 - 5x) = 7

3. Solve for the remaining variable: Simplify and solve for x: 2x + 27 - 15x = 7 -13x = -20 x = 20/13

4. Substitute back: Substitute the value of x (20/13) back into either of the original equations to solve for y. Let's use y = 9 - 5x: y = 9 - 5(20/13) y = 9 - 100/13 y = (117 - 100)/13 y = 17/13

Therefore, the solution to the system using the substitution method is x = 20/13 and y = 17/13.

2. The Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations to eliminate one variable by adding or subtracting the equations.

1. Multiply to create opposites: To eliminate y, we can multiply the second equation (5x + y = 9) by -3: -15x - 3y = -27

2. Add the equations: Now, add this modified equation to the first equation (2x + 3y = 7): 2x + 3y = 7 -15x - 3y = -27

   -13x = -20

3. Solve for x: Solve for x: x = 20/13

4. Substitute back: Substitute the value of x back into either of the original equations to solve for y. The result will be the same as the substitution method: y = 17/13

Therefore, the solution to the system using the elimination method is also x = 20/13 and y = 17/13.

3. The Graphical Method

The graphical method involves plotting both equations on a Cartesian plane. The point of intersection of the two lines represents the solution to the system. While this method is visually intuitive, it can be less precise than algebraic methods, especially when dealing with fractional solutions. In our case, the point of intersection would be (20/13, 17/13).

Verification of the Solution

To verify the solution (x = 20/13, y = 17/13), we substitute these values back into both original equations:

Equation 1: 2x + 3y = 7 2(20/13) + 3(17/13) = 40/13 + 51/13 = 91/13 = 7 (Correct)

Equation 2: 5x + y = 9 5(20/13) + 17/13 = 100/13 + 17/13 = 117/13 = 9 (Correct)

Both equations are satisfied, confirming the accuracy of our solution.

Significance of the Solution and Applications

The solution to this system of equations, (20/13, 17/13), represents a single unique point where the two lines intersect. This signifies that there is only one set of values for x and y that satisfies both equations simultaneously. This type of system is known as a consistent and independent system. Other types of systems include inconsistent systems (no solution, parallel lines) and dependent systems (infinite solutions, overlapping lines).

The applications of solving systems of linear equations are vast and span numerous fields:

• Engineering: Analyzing circuits, structural mechanics, and fluid dynamics often involves solving systems of linear equations.

• Economics: Linear programming, used in optimization problems like resource allocation and production planning, heavily relies on solving systems of equations.

• Computer Science: Computer graphics, machine learning algorithms, and simulations frequently utilize linear algebra and the solution of systems of equations.

• Physics: Many physical phenomena, like motion and forces, can be modeled using linear equations, requiring the solution of systems to analyze the behavior of the system.

Beyond the Basics: Matrix Representation and Solving Methods

For larger systems of linear equations (more than two variables), more advanced techniques become necessary. One such method involves using matrices. Our system can be represented in matrix form as:

[ 2  3 ] [ x ] = [ 7 ]
[ 5  1 ] [ y ]   [ 9 ]

This can be solved using techniques like Gaussian elimination, Cramer's rule, or matrix inversion. These methods are computationally efficient for solving larger systems and are implemented in various mathematical software packages.

Conclusion

Solving the system of linear equations 2x + 3y = 7 and 5x + y = 9 provides a practical illustration of fundamental mathematical concepts and their applications. We explored three different methods – substitution, elimination, and graphing – all leading to the same unique solution: x = 20/13 and y = 17/13. This seemingly simple problem highlights the power and versatility of linear algebra in tackling complex real-world problems across diverse disciplines. Understanding these methods is crucial for anyone pursuing studies in mathematics, science, engineering, or any field involving quantitative analysis. The ability to solve systems of equations efficiently and accurately is a fundamental skill with far-reaching implications.