Solving Systems Of Equations Step By Step Guide

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This article delves into the methods for solving a system of equations, providing a comprehensive understanding of the techniques involved. We will focus on solving the specific system presented, while also generalizing the approaches for broader applicability. The system of equations we aim to solve is:

$ \begin{array}{l} x-2 y=3 \ 2 x-3 y=9 \end{array} $

We will explore methods such as substitution and elimination to arrive at the solution, emphasizing the underlying principles and algebraic manipulations involved. Let's begin by understanding why solving systems of equations is so important.

Why Solve Systems of Equations?

Solving systems of equations is a fundamental skill in mathematics with widespread applications in various fields. Systems of equations arise when we have multiple variables and multiple equations relating these variables. The goal is to find values for the variables that satisfy all equations simultaneously. This is crucial in:

  • Science and Engineering: Modeling physical systems, analyzing circuits, solving for forces in mechanics, and many other applications often involve systems of equations.
  • Economics: Determining market equilibrium, analyzing supply and demand, and modeling economic growth often rely on solving systems of equations.
  • Computer Science: Solving linear systems is a core operation in areas like computer graphics, machine learning, and optimization algorithms.
  • Everyday Life: Even in simple scenarios like determining the cost of individual items when given a total price for a combination, systems of equations can be used.

Understanding the techniques to solve these systems provides a powerful tool for tackling real-world problems. In this article, we will explore two primary methods: substitution and elimination.

Method 1: The Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation in one variable, which can then be easily solved. Let's apply this to our system:

$ \begin{array}{l} x-2 y=3 \ 2 x-3 y=9 \end{array} $

  1. Solve for x in the first equation:

    Starting with the first equation, x−2y=3x - 2y = 3, we can isolate x by adding 2y to both sides:

    x=3+2yx = 3 + 2y

  2. Substitute into the second equation:

    Now we substitute this expression for x into the second equation, 2x−3y=92x - 3y = 9:

    2(3+2y)−3y=92(3 + 2y) - 3y = 9

  3. Solve for y:

    Expanding and simplifying the equation, we get:

    6+4y−3y=96 + 4y - 3y = 9

    y=3y = 3

  4. Substitute y back to find x:

    Now that we have the value of y, we can substitute it back into either of the original equations or the expression for x we derived earlier. Let's use x=3+2yx = 3 + 2y:

    x=3+2(3)x = 3 + 2(3)

    x=3+6x = 3 + 6

    x=9x = 9

Therefore, using the substitution method, we find that x = 9 and y = 3. This gives us the solution (9, 3).

Method 2: The Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations so that the coefficients of one of the variables are opposites. Then, by adding the equations, that variable is eliminated, and we can solve for the remaining variable. Let's apply this to our system:

$ \begin{array}{l} x-2 y=3 \ 2 x-3 y=9 \end{array} $

  1. Multiply equations to get opposite coefficients:

    To eliminate x, we can multiply the first equation by -2:

    −2(x−2y)=−2(3)-2(x - 2y) = -2(3)

    −2x+4y=−6-2x + 4y = -6

    Now we have the system:

    $ \begin{array}{l} -2x + 4y = -6 \ 2 x-3 y=9 \end{array} $

  2. Add the equations:

    Adding the two equations together, we eliminate x:

    (−2x+4y)+(2x−3y)=−6+9(-2x + 4y) + (2x - 3y) = -6 + 9

    y=3y = 3

  3. Substitute back to find x:

    Substitute the value of y back into one of the original equations. Let's use the first equation, x−2y=3x - 2y = 3:

    x−2(3)=3x - 2(3) = 3

    x−6=3x - 6 = 3

    x=9x = 9

Again, using the elimination method, we find the solution x = 9 and y = 3, which gives us the solution (9, 3).

Verifying the Solution

It's always a good practice to verify the solution by substituting the values of x and y back into the original equations. This ensures that the solution satisfies both equations.

  1. Substitute into the first equation:

    x−2y=3x - 2y = 3

    9−2(3)=39 - 2(3) = 3

    9−6=39 - 6 = 3

    3=33 = 3 (The first equation is satisfied)

  2. Substitute into the second equation:

    2x−3y=92x - 3y = 9

    2(9)−3(3)=92(9) - 3(3) = 9

    18−9=918 - 9 = 9

    9=99 = 9 (The second equation is satisfied)

Since the values x = 9 and y = 3 satisfy both equations, we can confidently say that (9, 3) is the correct solution to the system.

Comparing the Methods

Both the substitution and elimination methods are effective for solving systems of equations, but one method might be more convenient than the other depending on the specific system.

  • The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the other.
  • The elimination method is often preferred when the coefficients of one of the variables are already opposites or can be easily made opposites by multiplying one or both equations by a constant.

In this particular case, both methods work well. The substitution method required a straightforward substitution after solving for x, while the elimination method involved a single multiplication to create opposite coefficients. The choice of method often comes down to personal preference and recognizing the structure of the equations.

Common Mistakes to Avoid

When solving systems of equations, it's easy to make mistakes. Here are a few common pitfalls to avoid:

  • Arithmetic Errors: Be careful with arithmetic, especially when multiplying and adding equations. Double-check your calculations to avoid simple mistakes that can lead to incorrect solutions.
  • Incorrect Substitution: When using substitution, ensure you substitute the expression into the correct equation and variable. Substituting into the same equation you solved for can lead to a circular argument.
  • Sign Errors: Pay close attention to signs, especially when multiplying equations by negative numbers or distributing terms. A sign error can completely change the outcome.
  • Forgetting to Solve for Both Variables: Remember that the goal is to find the values of all variables in the system. Once you've solved for one variable, don't forget to substitute back to find the value of the other(s).

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving systems of equations.

Generalizing the Approach

The techniques we've used to solve this particular system can be generalized to solve other systems of linear equations. Here are some key takeaways:

  • Consistent Systems: A system of equations is consistent if it has at least one solution. The system we solved is consistent and has a unique solution.
  • Inconsistent Systems: A system is inconsistent if it has no solutions. This can occur if the equations represent parallel lines that never intersect.
  • Dependent Systems: A system is dependent if it has infinitely many solutions. This happens when the equations represent the same line or plane.
  • Systems with More Equations and Variables: The methods of substitution and elimination can be extended to systems with more than two equations and variables, although the calculations become more complex.

Understanding these concepts allows you to approach a wide range of systems of equations effectively. The principles of substitution and elimination remain fundamental, even as the complexity of the systems increases.

Conclusion

In this article, we've demonstrated how to solve a system of linear equations using both the substitution and elimination methods. We solved the system:

$ \begin{array}{l} x-2 y=3 \ 2 x-3 y=9 \end{array} $

and found the solution to be (9, 3). We also discussed the importance of solving systems of equations, the advantages and disadvantages of each method, common mistakes to avoid, and how to generalize these techniques to other systems. Mastering these skills provides a solid foundation for further study in mathematics and its applications.

The correct answer to the given problem is A. (9,3).