Solving Systems Of Equations A Comprehensive Guide To 18x + 5y = -11 And 12x + 11y = 31

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In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering and physics to economics and computer science. This article delves into a comprehensive approach to solving the system of equations: 18x + 5y = -11 and 12x + 11y = 31. We'll explore different methods, including substitution, elimination, and matrix operations, providing a step-by-step guide to each technique. By mastering these methods, you'll gain the ability to tackle a wide range of mathematical problems and real-world scenarios involving multiple variables and constraints.

Understanding Systems of Equations

Before diving into the solution, let's establish a clear understanding of what constitutes a system of equations. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. In other words, it's the point (or points) where the graphs of the equations intersect.

In our case, we have two linear equations with two variables (x and y). A linear equation represents a straight line on a graph, and the solution to the system will be the point where these two lines intersect. There are three possible scenarios for a system of two linear equations:

  1. Unique Solution: The lines intersect at a single point, indicating a unique solution for x and y.
  2. No Solution: The lines are parallel and never intersect, meaning there is no solution that satisfies both equations.
  3. Infinite Solutions: The lines are coincident (they overlap), indicating an infinite number of solutions.

Our goal is to determine which of these scenarios applies to our system and, if a unique solution exists, to find the values of x and y.

Method 1 The Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This eliminates one variable, allowing us to solve for the remaining variable. Let's apply this method to our system:

  1. Choose an equation and solve for one variable: Let's choose the first equation, 18x + 5y = -11, and solve for y: 5y = -11 - 18x y = (-11 - 18x) / 5

  2. Substitute the expression into the other equation: Substitute the expression for y into the second equation, 12x + 11y = 31: 12x + 11((-11 - 18x) / 5) = 31

  3. Solve for the remaining variable: Simplify and solve for x: 12x - (121 + 198x) / 5 = 31 Multiply both sides by 5 to eliminate the fraction: 60x - 121 - 198x = 155 Combine like terms: -138x = 276 Divide both sides by -138: x = -2

  4. Substitute the value back to find the other variable: Substitute x = -2 into the expression for y: y = (-11 - 18(-2)) / 5 y = (-11 + 36) / 5 y = 25 / 5 y = 5

Therefore, using the substitution method, we find that the solution to the system of equations is x = -2 and y = 5.

Method 2 The Elimination Method

The elimination method, also known as the addition or subtraction method, involves manipulating the equations so that the coefficients of one variable are opposites. When the equations are added, that variable is eliminated, allowing us to solve for the remaining variable. Let's apply this method to our system:

  1. Multiply the equations to make the coefficients of one variable opposites: Let's eliminate x. Multiply the first equation (18x + 5y = -11) by 2 and the second equation (12x + 11y = 31) by -3: 36x + 10y = -22 -36x - 33y = -93

  2. Add the equations to eliminate one variable: Add the two equations together: (36x + 10y) + (-36x - 33y) = -22 + (-93) -23y = -115

  3. Solve for the remaining variable: Divide both sides by -23: y = 5

  4. Substitute the value back to find the other variable: Substitute y = 5 into either of the original equations. Let's use the first equation, 18x + 5y = -11: 18x + 5(5) = -11 18x + 25 = -11 18x = -36 x = -2

Using the elimination method, we arrive at the same solution: x = -2 and y = 5. This confirms the consistency of our results.

Method 3 Matrix Operations

Matrix operations provide a powerful and efficient way to solve systems of linear equations, especially for larger systems with more variables. This method involves representing the system in matrix form and using techniques like Gaussian elimination or matrix inversion. Let's apply this method to our system.

  1. Represent the system in matrix form: Our system of equations can be written in matrix form as Ax = b, where: A is the coefficient matrix:

    [[18, 5],
 [12, 11]]

    x is the variable matrix:

    [[x],
 [y]]

    b is the constant matrix:

    [[-11],
 [31]]

  2. Use Gaussian elimination or matrix inversion to solve for x: We'll use Gaussian elimination, which involves transforming the augmented matrix [A | b] into row-echelon form. The augmented matrix is:

    [[18, 5, -11],
 [12, 11, 31]]

    Perform row operations to get a leading 1 in the first row, first column: Divide the first row by 18:

    [[1, 5/18, -11/18],
 [12, 11, 31]]

    Eliminate the 12 in the second row, first column: Subtract 12 times the first row from the second row:

    [[1, 5/18, -11/18],
 [0, 167/18, 674/18]]

    Get a leading 1 in the second row, second column: Multiply the second row by 18/167:

    [[1, 5/18, -11/18],
 [0, 1, 674/167]]

    Simplify 674/167 = 4044/1002 = 2022/501

    [[1, 5/18, -11/18],
 [0, 1, 5]]

    Eliminate the 5/18 in the first row, second column: Subtract 5/18 times the second row from the first row:

    [[1, 0, -2],
 [0, 1, 5]]

    The matrix is now in row-echelon form. The solution is x = -2 and y = 5.

Using matrix operations, we once again arrive at the solution x = -2 and y = 5, further validating our results.

Verification and Conclusion

To ensure the accuracy of our solution, let's substitute the values x = -2 and y = 5 back into the original equations:

18x + 5y = -11: 18(-2) + 5(5) = -36 + 25 = -11 (Correct)

12x + 11y = 31: 12(-2) + 11(5) = -24 + 55 = 31 (Correct)

The values satisfy both equations, confirming that our solution is correct.

In conclusion, we've successfully solved the system of equations 18x + 5y = -11 and 12x + 11y = 31 using three different methods: substitution, elimination, and matrix operations. All methods yielded the same solution: x = -2 and y = 5. This demonstrates the versatility of these techniques and provides a comprehensive understanding of how to approach and solve systems of linear equations. Mastering these methods is crucial for various mathematical and real-world applications, empowering you to tackle complex problems with confidence.

This exploration highlights the importance of understanding different problem-solving approaches in mathematics. Each method offers a unique perspective and can be more efficient depending on the specific system of equations. By mastering these techniques, you'll be well-equipped to handle a wide range of mathematical challenges. Remember, practice is key to proficiency, so continue to explore different systems of equations and apply these methods to solidify your understanding. This will not only enhance your mathematical skills but also sharpen your critical thinking and problem-solving abilities, valuable assets in any field. The ability to solve systems of equations is a cornerstone of mathematical literacy, and with consistent effort, you can master this essential skill.

In addition to the methods discussed, there are other techniques for solving systems of equations, such as graphing. Graphing the equations and finding the point of intersection provides a visual representation of the solution. This method can be particularly useful for understanding the nature of the solutions and identifying cases where there are no solutions or infinite solutions. Furthermore, software and online calculators can assist in solving complex systems of equations, allowing for efficient computation and verification of results. However, it's crucial to understand the underlying principles and methods to effectively utilize these tools and interpret the results accurately.

Solving systems of equations is not just a theoretical exercise; it has practical applications in numerous fields. In engineering, it's used to analyze circuits, design structures, and optimize processes. In economics, it helps model supply and demand, predict market trends, and allocate resources. In computer science, it's applied in areas like computer graphics, artificial intelligence, and cryptography. By developing a strong foundation in solving systems of equations, you'll unlock opportunities to apply mathematical concepts to real-world problems and contribute to various industries. The power of mathematics lies in its ability to model and solve complex problems, and mastering systems of equations is a significant step towards harnessing that power.