8x 5y 11 4x 3y 5

Decoding the Mystery: A Deep Dive into the Equation 8x + 5y = 11 and 4x + 3y = 5

This article delves into the mathematical puzzle presented by the simultaneous equations 8x + 5y = 11 and 4x + 3y = 5. We'll explore various methods for solving these equations, examining their graphical representation, and discussing the implications of their solutions. We'll also touch upon the broader mathematical concepts involved and how these types of problems are applied in real-world scenarios.

Understanding Simultaneous Equations

Simultaneous equations, also known as systems of equations, involve two or more equations with two or more variables. The goal is to find the values of the variables that satisfy all equations simultaneously. In our case, we have two linear equations:

• 8x + 5y = 11
• 4x + 3y = 5

These equations represent straight lines on a graph. The solution to the simultaneous equations is the point where these two lines intersect.

Method 1: Elimination Method

The elimination method aims to eliminate one variable by manipulating the equations. Let's eliminate 'x' from our equations:

1. Multiply the second equation by -2: This gives us -8x - 6y = -10.

2. Add this modified equation to the first equation: (8x + 5y) + (-8x - 6y) = 11 + (-10). This simplifies to -y = 1, or y = -1.

3. Substitute y = -1 into either of the original equations to solve for x. Let's use the second equation: 4x + 3(-1) = 5. This simplifies to 4x - 3 = 5, so 4x = 8, and x = 2.

Therefore, the solution to the simultaneous equations is x = 2 and y = -1.

Method 2: Substitution Method

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

1. Solve the second equation for x: 4x = 5 - 3y => x = (5 - 3y) / 4

2. Substitute this expression for x into the first equation: 8((5 - 3y) / 4) + 5y = 11

3. Simplify and solve for y: 2(5 - 3y) + 5y = 11 => 10 - 6y + 5y = 11 => -y = 1 => y = -1

4. Substitute y = -1 back into either of the original equations to solve for x. Using the second equation again: 4x + 3(-1) = 5 => 4x = 8 => x = 2

Again, we arrive at the solution x = 2 and y = -1.

Method 3: Graphical Method

The graphical method involves plotting both equations on a graph. The point of intersection represents the solution.

To plot the lines, we can find at least two points for each equation:

Equation 1: 8x + 5y = 11

• If x = 0, then 5y = 11, so y = 11/5 = 2.2
• If y = 0, then 8x = 11, so x = 11/8 = 1.375

Equation 2: 4x + 3y = 5

• If x = 0, then 3y = 5, so y = 5/3 ≈ 1.67
• If y = 0, then 4x = 5, so x = 5/4 = 1.25

By plotting these points and drawing the lines, the intersection point will be at approximately x = 2 and y = -1. This method is less precise than algebraic methods but provides a visual representation of the solution.

Verifying the Solution

To ensure our solution is correct, we substitute x = 2 and y = -1 into both original equations:

Equation 1: 8(2) + 5(-1) = 16 - 5 = 11 (Correct)

Equation 2: 4(2) + 3(-1) = 8 - 3 = 5 (Correct)

Both equations are satisfied, confirming that x = 2 and y = -1 is the correct solution.

Applications of Simultaneous Equations

Simultaneous equations have wide-ranging applications in various fields:

• Physics: Solving problems involving forces, motion, and electricity. For example, determining the speed and direction of two objects moving towards each other.

• Engineering: Analyzing circuits, designing structures, and solving problems related to fluid dynamics. This involves complex systems of equations often solved with computers.

• Economics: Modeling supply and demand, analyzing market equilibrium, and forecasting economic trends.

• Chemistry: Determining the concentration of substances in mixtures, balancing chemical equations, and analyzing reaction kinetics.

• Computer Graphics: Used in 3D modeling and animation to transform and manipulate objects in space.

Expanding the Problem: Variations and Extensions

The problem presented, while seemingly simple, forms the foundation for understanding more complex systems of equations. Consider these extensions:

• Non-linear equations: Instead of straight lines, the equations could represent curves (parabolas, circles, etc.), leading to multiple solutions or no solutions.

• Systems with more than two variables: Solving three or more equations with three or more variables requires more sophisticated techniques like matrix methods (Gaussian elimination, Cramer's rule).

• Inequalities: Instead of equalities, the problem could involve inequalities (e.g., 8x + 5y > 11), leading to regions of solutions rather than single points.

Conclusion

Solving the simultaneous equations 8x + 5y = 11 and 4x + 3y = 5 demonstrates the fundamental principles of solving linear systems. We explored three distinct methods – elimination, substitution, and graphical – each offering a unique approach to finding the solution (x = 2, y = -1). The verification process ensures the accuracy of the result. Understanding these methods is crucial for tackling more complex mathematical problems across various scientific and engineering disciplines. The seemingly simple problem provides a strong base for grasping the power and applicability of simultaneous equations in a wide range of real-world scenarios. Further exploration into non-linear systems and systems with more variables will significantly expand your mathematical capabilities and problem-solving skills.