7x 2y 5 5x 9y 1

Decoding the Mystery: A Deep Dive into 7x + 2y = 5 and 5x + 9y = 1

This article delves into the intricacies of solving a system of two linear equations with two variables: 7x + 2y = 5 and 5x + 9y = 1. We'll explore multiple methods for finding the solution, discussing their strengths and weaknesses, and providing a comprehensive understanding of the underlying mathematical principles. Beyond the solution itself, we will also explore the geometrical interpretation of these equations and the significance of the solution in the context of linear algebra.

Understanding Linear Equations

Before we embark on solving the system, let's briefly refresh our understanding of linear equations. A linear equation is an algebraic equation of the form ax + by = c, where 'a', 'b', and 'c' are constants, and 'x' and 'y' are variables. The graph of a linear equation is always a straight line. The solution to a linear equation represents the coordinates (x, y) of points that lie on that line.

Our system of equations, 7x + 2y = 5 and 5x + 9y = 1, involves two such linear equations. Finding a solution means discovering the values of 'x' and 'y' that satisfy both equations simultaneously. Geometrically, this means finding the point of intersection of the two lines represented by these equations.

Method 1: Elimination Method

The elimination method, also known as the addition method, is a classic approach to solving systems of linear equations. The goal is to eliminate one variable by manipulating the equations to create opposite coefficients for that variable. Let's apply this to our system:

1. Multiply the equations to create opposite coefficients: We can multiply the first equation by 9 and the second equation by -2 to eliminate 'y':

   • 9(7x + 2y) = 9(5) => 63x + 18y = 45
   • -2(5x + 9y) = -2(1) => -10x - 18y = -2

2. Add the modified equations: Adding the two modified equations eliminates 'y':

   • (63x + 18y) + (-10x - 18y) = 45 + (-2)
   • 53x = 43

3. Solve for x:

   • x = 43/53

4. Substitute x back into either original equation to solve for y: Let's use the first equation:

   • 7(43/53) + 2y = 5
   • 301/53 + 2y = 5
   • 2y = 5 - 301/53
   • 2y = (265 - 301)/53
   • 2y = -36/53
   • y = -18/53

Therefore, the solution using the elimination method is x = 43/53 and y = -18/53.

Method 2: 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 one equation for one variable: Let's solve the first equation for x:

   • 7x + 2y = 5
   • 7x = 5 - 2y
   • x = (5 - 2y)/7

2. Substitute this expression for x into the second equation:

   • 5((5 - 2y)/7) + 9y = 1
   • (25 - 10y)/7 + 9y = 1
   • 25 - 10y + 63y = 7
   • 53y = -18
   • y = -18/53

3. Substitute the value of y back into either original equation to solve for x: Using the first equation again:

   • 7x + 2(-18/53) = 5
   • 7x - 36/53 = 5
   • 7x = 5 + 36/53
   • 7x = (265 + 36)/53
   • 7x = 301/53
   • x = 43/53

Again, the solution is x = 43/53 and y = -18/53.

Method 3: Using Matrices and Cramer's Rule

For those familiar with linear algebra, Cramer's Rule provides an elegant method for solving systems of linear equations using matrices. Let's represent our system in matrix form:

| 7  2 |   | x |   | 5 |
| 5  9 | * | y | = | 1 |

1. Calculate the determinant of the coefficient matrix:

   • D = (7 * 9) - (2 * 5) = 63 - 10 = 53

2. Calculate the determinant of the matrix formed by replacing the x-column with the constants:

   • Dx = (5 * 9) - (2 * 1) = 45 - 2 = 43

3. Calculate the determinant of the matrix formed by replacing the y-column with the constants:

   • Dy = (7 * 1) - (5 * 5) = 7 - 25 = -18

4. Apply Cramer's Rule to find x and y:

   • x = Dx / D = 43/53
   • y = Dy / D = -18/53

Once again, we arrive at the solution x = 43/53 and y = -18/53.

Geometrical Interpretation

Geometrically, each equation represents a straight line on a Cartesian plane. The solution (x = 43/53, y = -18/53) represents the point of intersection of these two lines. This point is the only coordinate pair that satisfies both equations simultaneously. If the lines were parallel, there would be no solution (inconsistent system). If the lines were coincident (identical), there would be infinitely many solutions (dependent system). In our case, the lines intersect at a unique point, indicating a consistent and independent system.

Applications and Significance

Systems of linear equations have widespread applications across various fields:

• Engineering: Analyzing circuits, determining stresses in structures, and modeling dynamic systems.
• Physics: Solving problems involving forces, motion, and energy.
• Economics: Modeling supply and demand, optimizing resource allocation, and forecasting economic trends.
• Computer Science: Solving problems in computer graphics, optimization algorithms, and machine learning.
• Business and Finance: Analyzing financial models, forecasting sales, and managing portfolios.

The ability to efficiently and accurately solve these systems is crucial for tackling real-world problems in these and many other domains. The methods discussed in this article provide a robust foundation for understanding and solving such problems.

Conclusion

Solving the system of equations 7x + 2y = 5 and 5x + 9y = 1, whether through elimination, substitution, or Cramer's Rule, yields the unique solution x = 43/53 and y = -18/53. Understanding these solution methods and their geometrical interpretations is essential for anyone working with linear equations and their diverse applications. This article has aimed to provide a thorough and accessible explanation of the process, reinforcing the fundamental principles of linear algebra and its practical relevance. The significance of mastering these techniques extends far beyond simple equation solving, offering a cornerstone for more complex mathematical modeling and problem-solving endeavors. Remember to practice these methods with various examples to solidify your understanding and develop your problem-solving skills.