7x 2y 5 5x 9y 1

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

This article delves into the mathematical intricacies of solving a system of two linear equations with two variables: 7x + 2y = 5 and 5x + 9y = 1. We'll explore various methods to find the solution, discuss the significance of such equations in real-world applications, and offer insights into the underlying mathematical concepts. Our goal is not just to provide the answer, but to empower you with a thorough understanding of the process.

Understanding Linear Equations and Systems of Equations

Before we tackle the specific problem, let's establish a solid foundation. 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. Graphically, a linear equation represents a straight line.

A system of linear equations involves two or more linear equations with the same variables. The solution to a system of equations is the set of values for the variables that satisfy all the equations simultaneously. Geometrically, this represents the point(s) of intersection between the lines.

Our system, 7x + 2y = 5 and 5x + 9y = 1, is a system of two linear equations with two variables (x and y). Finding the solution means determining the values of x and y that make both equations true.

Methods for Solving Systems of Linear Equations

Several methods exist for solving systems of linear equations. We'll explore three common techniques:

1. Elimination Method (or Addition Method)

The elimination method aims to eliminate one variable by adding or subtracting the equations. To eliminate a variable, we need to make the coefficients of that variable opposites.

Let's apply this to our system:

• Equation 1: 7x + 2y = 5
• Equation 2: 5x + 9y = 1

We can multiply Equation 1 by 9 and Equation 2 by -2 to eliminate 'y':

• Modified Equation 1: 63x + 18y = 45
• Modified Equation 2: -10x - 18y = -2

Now, add the modified equations:

53x = 43

Solving for x:

x = 43/53

Substitute this value of x into either original equation to solve for y. Let's use Equation 1:

7(43/53) + 2y = 5

Solving for y:

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.

2. Substitution Method

The substitution method involves solving one equation for one variable and substituting the expression into the other equation.

Let's solve Equation 1 for x:

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

Now, substitute this expression for x into Equation 2:

5((5 - 2y)/7) + 9y = 1

Solving for y:

25 - 10y + 63y = 7 53y = -18 y = -18/53

Substitute this value of y back into the expression for x:

x = (5 - 2(-18/53))/7 x = (5 + 36/53)/7 x = (265 + 36)/371 x = 301/371 x = 43/53

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

3. Graphical Method

The graphical method involves plotting both equations on a graph. The point where the two lines intersect represents the solution. While this method is visually intuitive, it can be less precise than algebraic methods, especially when dealing with fractions or decimals. For our equations, the intersection point would be approximately (0.81, -0.34). This illustrates the approximate nature of this method when compared to the precise solutions obtained via elimination or substitution.

Verification of the Solution

It's crucial to verify the solution by substituting the values of x and y into both original equations:

Equation 1: 7(43/53) + 2(-18/53) = 301/53 - 36/53 = 265/53 = 5 (Correct)

Equation 2: 5(43/53) + 9(-18/53) = 215/53 - 162/53 = 53/53 = 1 (Correct)

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

Real-World Applications of Systems of Linear Equations

Systems of linear equations are not merely abstract mathematical concepts; they have extensive real-world applications across various fields:

• Economics: Analyzing supply and demand, optimizing production, and modeling economic relationships.
• Engineering: Solving circuit problems, analyzing structural mechanics, and designing control systems.
• Physics: Modeling motion, calculating forces, and analyzing electrical circuits.
• Computer Science: Developing algorithms, solving optimization problems, and creating simulations.
• Finance: Portfolio management, risk assessment, and determining optimal investment strategies.

The ability to solve systems of linear equations is a fundamental skill in many quantitative fields.

Advanced Concepts and Extensions

The system we analyzed is a simple example. More complex scenarios involve:

• More than two variables: Systems with three or more variables require more sophisticated techniques like Gaussian elimination or matrix methods.
• Non-linear equations: Equations that do not form straight lines when graphed require different approaches like substitution or numerical methods.
• Systems with no solutions or infinitely many solutions: Some systems of equations may have no common solution (parallel lines) or infinitely many solutions (overlapping lines).

Understanding these more advanced concepts is crucial for tackling more challenging mathematical problems.

Conclusion

Solving the system of equations 7x + 2y = 5 and 5x + 9y = 1 provided us with an opportunity to explore different methods for solving systems of linear equations. The elimination, substitution, and graphical methods each offer unique advantages and provide valuable insights into the underlying mathematical principles. The solution, x = 43/53 and y = -18/53, was verified, demonstrating the importance of accurate calculation and verification in problem-solving. Furthermore, we've highlighted the wide-ranging applicability of these seemingly simple equations in diverse fields, underscoring their significance beyond the realm of pure mathematics. Mastering these techniques is essential for anyone pursuing studies or careers involving quantitative analysis and problem-solving. The exploration of these concepts provides a strong foundation for tackling more complex mathematical challenges in the future.