7x 2y 24 8x 2y 30

Deciphering the Enigma: Solving the Simultaneous Equations 7x + 2y = 24 and 8x + 2y = 30

This article delves into the solution of the simultaneous equations 7x + 2y = 24 and 8x + 2y = 30. We'll explore various methods to solve these equations, discuss their applications in real-world scenarios, and offer insights into the broader mathematical concepts involved. This comprehensive guide will cater to students, mathematicians, and anyone curious about the elegant world of simultaneous equations.

Understanding Simultaneous Equations

Simultaneous equations, also known as systems of equations, are a set of two or more equations that contain two or more variables. The goal is to find the values of the variables that satisfy all the equations simultaneously. In our case, we have two linear equations with two variables, x and y. This type of system is relatively straightforward to solve using various techniques.

Why are Simultaneous Equations Important?

Simultaneous equations are fundamental to many areas, including:

• Physics: Modeling motion, forces, and circuits.
• Engineering: Designing structures, analyzing systems, and optimizing processes.
• Economics: Analyzing market equilibrium, supply and demand, and economic models.
• Computer Science: Solving linear systems in algorithms and simulations.
• Chemistry: Calculating concentrations and reaction rates.

Methods for Solving Simultaneous Equations

Several methods can be used to solve simultaneous equations. We'll explore three common approaches: elimination, substitution, and graphical representation.

1. Elimination Method

The elimination method involves manipulating the equations to eliminate one variable, leaving a single equation with one variable that can be easily solved. Let's apply this to our equations:

7x + 2y = 24 (Equation 1) 8x + 2y = 30 (Equation 2)

Notice that the '2y' term is identical in both equations. Subtracting Equation 1 from Equation 2 will eliminate 'y':

(8x + 2y) - (7x + 2y) = 30 - 24 8x - 7x + 2y - 2y = 6 x = 6

Now that we have the value of x, we can substitute it back into either Equation 1 or Equation 2 to solve for y. Let's use Equation 1:

7(6) + 2y = 24 42 + 2y = 24 2y = 24 - 42 2y = -18 y = -9

Therefore, the solution to the simultaneous equations is x = 6 and y = -9.

2. Substitution Method

The substitution method involves solving one equation for one variable in terms of the other and substituting this expression into the second equation.

Let's solve Equation 1 for y:

7x + 2y = 24 2y = 24 - 7x y = 12 - (7/2)x

Now, substitute this expression for y into Equation 2:

8x + 2(12 - (7/2)x) = 30 8x + 24 - 7x = 30 x = 30 - 24 x = 6

Substitute the value of x (6) back into the expression for y:

y = 12 - (7/2)(6) y = 12 - 21 y = -9

Again, we arrive at the solution x = 6 and y = -9.

3. Graphical Representation

Simultaneous equations can also be solved graphically. Each equation represents a straight line on a Cartesian plane. The point where the two lines intersect represents the solution to the system of equations.

To graph these equations, we can find two points on each line and draw a line through them. For example, for 7x + 2y = 24:

• If x = 0, then 2y = 24, so y = 12. One point is (0, 12).
• If y = 0, then 7x = 24, so x = 24/7 ≈ 3.43. Another point is (24/7, 0).

Similarly, for 8x + 2y = 30:

• If x = 0, then 2y = 30, so y = 15. One point is (0, 15).
• If y = 0, then 8x = 30, so x = 30/8 = 15/4 = 3.75. Another point is (15/4, 0).

Plotting these points and drawing the lines, you'll find that they intersect at the point (6, -9), confirming our solution. However, this method can be less precise than algebraic methods, especially if the solution involves fractional or decimal values.

Verifying the Solution

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

Equation 1: 7(6) + 2(-9) = 42 - 18 = 24 (Correct) Equation 2: 8(6) + 2(-9) = 48 - 18 = 30 (Correct)

The solution x = 6, y = -9 satisfies both equations, confirming its accuracy.

Applications of Simultaneous Equations

The applications of simultaneous equations extend far beyond simple mathematical exercises. Consider these examples:

• Mixture Problems: Suppose you need to mix two solutions with different concentrations to obtain a specific concentration. Simultaneous equations can help determine the quantities of each solution needed.
• Rate Problems: If you have two objects moving at different speeds and want to find the time when they meet or the distance between them at a particular time, simultaneous equations are invaluable.
• Cost and Revenue Analysis: In business, simultaneous equations can model cost and revenue functions to determine break-even points or profit maximization.
• Network Analysis: In computer networks or traffic flow, simultaneous equations can be used to model traffic distribution or data flow.
• Linear Programming: Many optimization problems, such as resource allocation and scheduling, involve solving systems of linear equations and inequalities.

Advanced Concepts and Extensions

The principles discussed here can be extended to more complex systems of equations involving three or more variables. Matrix algebra provides a powerful tool for solving such systems efficiently. Furthermore, the concepts of linear dependence and independence are crucial in determining the solvability and uniqueness of solutions to simultaneous equations. These more advanced topics often form the basis of linear algebra courses.

Conclusion

Solving simultaneous equations, such as 7x + 2y = 24 and 8x + 2y = 30, is a fundamental skill in mathematics with far-reaching applications. Understanding the various methods – elimination, substitution, and graphical representation – provides a versatile toolkit for tackling these problems. By mastering these techniques, you'll be well-equipped to handle more complex mathematical challenges and apply these concepts to diverse real-world scenarios, strengthening your analytical and problem-solving skills. Remember to always verify your solutions to ensure accuracy and develop a deep understanding of the underlying principles involved.