7x 2y 24 8x 2y 30

Solving Systems of Linear Equations: AStep-by-Step Guide to 7x + 2y = 24 and 8x + 2y = 30

Understanding how to solve systems of linear equations is a fundamental skill in algebra with wide-ranging applications, from physics to economics. This guide provides a clear, step-by-step approach to solving the specific system presented by the equations 7x + 2y = 24 and 8x + 2y = 30. Mastering these techniques empowers you to find the unique values of the variables that satisfy both equations simultaneously.

Introduction

A system of linear equations consists of two or more equations that share the same variables. The goal is to find the values of those variables that make all equations true at the same time. The given system is:

1. 7x + 2y = 24
2. 8x + 2y = 30

Notice that both equations contain the term +2y. This shared term is key to a simple solution method. This article will walk you through the process of solving this system using the Elimination Method, explain the underlying principles, and address common questions. By the end, you'll understand how to find the values of x and y that solve both equations.

Steps to Solve the System

1. Identify the Equations: Clearly write down the system:

   • Equation 1: 7x + 2y = 24
   • Equation 2: 8x + 2y = 30

2. Look for a Common Term: Observe that both equations have +2y. This is crucial because it means the coefficient of y is identical in both equations.

3. Eliminate the Common Term (Subtraction): To eliminate y, subtract Equation 1 from Equation 2. This cancels out the 2y terms:

   • (8x + 2y) - (7x + 2y) = 30 - 24
   • This simplifies to: 8x + 2y - 7x - 2y = 6
   • Therefore: x = 6

4. Solve for the First Variable: You've found the value of x. It's 6.

5. Substitute to Find the Second Variable: Now that you know x = 6, plug this value into either original equation to solve for y. Using Equation 1:

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

6. Verify the Solution: Always check your solution by plugging x = 6 and y = -9 back into both original equations.

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

The solution to the system 7x + 2y = 24 and 8x + 2y = 30 is x = 6 and y = -9.

Scientific Explanation

The elimination method leverages the property that adding or subtracting equivalent equations (equations that represent the same relationship) produces another equation that is also true. In this case, subtracting Equation 1 from Equation 2 effectively removes the 2y term because they are identical and thus cancel each other out. This leaves us with a single equation in x (7x - 8x = x), which is much easier to solve. Once x is known, substitution provides the value of y by leveraging the relationship defined by either original equation. This process demonstrates the core principle of solving linear systems: manipulating equations to isolate variables.

Frequently Asked Questions (FAQ)

• Q: Why did subtracting the equations work to eliminate y?
  • A: Both equations have the same coefficient (2) and sign (+) for y. When you subtract one equation from the other, the y terms subtract out (2y - 2y = 0), leaving only the x terms.
• Q: Could I have used addition instead?
  • A: Yes, but it wouldn't eliminate y directly. You could multiply one equation by -1 and then add it to the other, but subtraction is simpler here.
• Q: What if the coefficients of y were different?
  • A: You would need to multiply one or both equations by a number to make the coefficients of y the same (or negatives) before subtracting. This is a key step in the general elimination method.
• Q: Is there only one solution?
  • A: For a consistent system with non-parallel lines (like this one), there is exactly one unique solution, represented by the point (6, -9). If the lines were parallel (same slope, different y-intercepts), there would be no solution. If they were identical, there would be infinitely many solutions.
• Q: Why is it important to verify the solution?
  • A: Verification ensures you didn't make an arithmetic error during the solving process. Plugging the values back into the original equations confirms they satisfy both conditions simultaneously.

Conclusion

Solving the system 7x + 2y = 24 and 8x + 2y = 30 is a straightforward application of the elimination method. By recognizing the shared term +2y, subtraction efficiently isolates x, leading to the solution x = 6 and y = -9. This process highlights the power of algebraic manipulation to find precise values that satisfy multiple constraints. Understanding these fundamental techniques is essential for tackling more complex problems in mathematics, science, engineering, and everyday problem-solving. Remember to always verify your solution to ensure accuracy.

This method, while effective for simple linear equations, forms the bedrock for tackling more intricate systems involving multiple variables and non-linear relationships. More advanced techniques like substitution, matrices, and graphical methods build upon these foundational principles. The elimination method's strength lies in its systematic approach to reducing complexity, making it a valuable tool in any mathematical toolkit. It teaches us to identify and exploit common factors and relationships within equations to efficiently arrive at a solution. Furthermore, the process of verification reinforces the importance of careful calculation and attention to detail – a skill transferable to countless areas beyond mathematics. The ability to translate real-world scenarios into mathematical equations and then solve them using these techniques is a powerful skill that empowers us to analyze, understand, and predict outcomes in a wide range of situations. Therefore, mastering the elimination method isn't just about solving equations; it's about developing a logical and analytical mindset crucial for success in a data-driven world.

The elimination method's systematic approach to reducing complexity makes it a valuable tool in any mathematical toolkit. It teaches us to identify and exploit common factors and relationships within equations to efficiently arrive at a solution. Furthermore, the process of verification reinforces the importance of careful calculation and attention to detail—a skill transferable to countless areas beyond mathematics. The ability to translate real-world scenarios into mathematical equations and then solve them using these techniques is a powerful skill that empowers us to analyze, understand, and predict outcomes in a wide range of situations. Therefore, mastering the elimination method isn't just about solving equations; it's about developing a logical and analytical mindset crucial for success in a data-driven world.

Beyond its direct application in solving for unknowns, the elimination method fosters a deeper understanding of equation equivalence. Manipulating equations through addition or subtraction doesn’t change the solution set; it merely presents the information in a different, often more accessible, form. This concept of maintaining equivalence is paramount in more advanced algebraic manipulations, such as solving inequalities or working with systems of non-linear equations. Consider, for instance, a scenario involving mixing chemical solutions where concentrations and volumes must satisfy specific relationships. Setting up a system of equations and employing the elimination method allows for precise determination of the required quantities, preventing errors and ensuring desired outcomes.

The method also provides a strong foundation for understanding concepts in linear algebra, where systems of equations are represented and solved using matrices. The row operations used in Gaussian elimination, a core technique in linear algebra, are directly analogous to the addition and subtraction operations used in the elimination method for two or three variables. This connection demonstrates the scalability of the underlying principles. Even in fields like economics, where supply and demand curves are often represented as linear equations, the elimination method can be used to find equilibrium points – the price and quantity where supply equals demand.

In conclusion, the elimination method is far more than a procedural technique for solving linear equations. It’s a cornerstone of algebraic thinking, cultivating analytical skills, reinforcing the importance of precision, and providing a gateway to more advanced mathematical concepts. Its applicability extends far beyond the classroom, empowering individuals to model and solve problems across a diverse spectrum of disciplines. By embracing this method and understanding its underlying principles, we equip ourselves with a powerful tool for navigating and interpreting the quantitative world around us.