Solving Systems Of Equations A Comprehensive Guide

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In the realm of mathematics, solving systems of equations is a fundamental skill with wide-ranging applications across various fields, from science and engineering to economics and computer science. A system of equations is a set of two or more equations that share the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. This article delves into the intricacies of solving systems of linear equations, providing a step-by-step guide and illustrating different methods with examples. We will focus on a specific system of equations and explore how to find its solution using various techniques.

Understanding Systems of Linear Equations

Before we dive into the methods, let's first understand what linear equations are. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variables are raised to the power of one, and there are no products or other functions of the variables. A system of linear equations, therefore, consists of two or more linear equations involving the same variables.

The system of equations presented is:

-3x + 6y = 9
5x + 7y = -49

This system involves two linear equations with two variables, x and y. The solution to this system is a pair of values (x, y) that satisfies both equations simultaneously. Geometrically, each equation represents a straight line in the coordinate plane, and the solution corresponds to the point where the lines intersect. If the lines are parallel, there is no solution, and if the lines coincide, there are infinitely many solutions.

Methods for Solving Systems of Linear Equations

There are several methods for solving systems of linear equations, including:

  1. Substitution Method: This method involves solving one equation for one variable and substituting the expression into the other equation. This results in a single equation with one variable, which can then be solved. The value of the first variable can then be substituted back into either of the original equations to find the value of the second variable.
  2. Elimination Method (or Addition Method): This method involves manipulating the equations so that the coefficients of one of the variables are opposites. Adding the equations then eliminates that variable, leaving a single equation with one variable. This equation can be solved, and the value can be substituted back into one of the original equations to find the value of the other variable.
  3. Graphical Method: This method involves graphing the equations on the coordinate plane. The solution to the system is the point of intersection of the lines. This method is particularly useful for visualizing the system and understanding the nature of the solutions.
  4. Matrix Method: This method involves representing the system of equations in matrix form and using matrix operations to solve for the variables. This method is especially efficient for systems with many equations and variables.

Let's explore the substitution and elimination methods to solve the given system of equations. Understanding these methods will equip you with the necessary tools to tackle various systems of equations.

1. Substitution Method: A Step-by-Step Approach

The substitution method is a powerful algebraic technique to solve systems of linear equations. In this method, we first isolate one variable in one of the equations. Then, we substitute the expression for that variable into the other equation. This substitution results in a single equation with only one variable, which we can easily solve. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. Let's apply this method to the system:

-3x + 6y = 9
5x + 7y = -49

Step 1: Solve one equation for one variable.

Let's choose the first equation, -3x + 6y = 9, and solve for x. To do this, we first isolate the term with x:

-3x = -6y + 9

Now, divide both sides by -3:

x = 2y - 3

Step 2: Substitute the expression into the other equation.

Next, substitute this expression for x (2y - 3) into the second equation, 5x + 7y = -49:

5(2y - 3) + 7y = -49

Step 3: Solve the resulting equation for the remaining variable.

Now we have an equation with only y. Let's solve for y:

10y - 15 + 7y = -49
17y - 15 = -49
17y = -34
y = -2

Step 4: Substitute the value back into one of the original equations to find the other variable.

Now that we have y = -2, substitute this value back into the expression we found for x in Step 1, x = 2y - 3:

x = 2(-2) - 3
x = -4 - 3
x = -7

Thus, the solution to the system of equations using the substitution method is x = -7 and y = -2. This means the point of intersection of the two lines represented by the equations is (-7, -2).

2. Elimination Method: Streamlining the Process

The elimination method, also known as the addition method, is another powerful technique for solving systems of linear equations. This method is particularly effective when the coefficients of one of the variables are either the same or easy to make the same (or opposites) by multiplying the equations by suitable constants. The key idea behind this method is to eliminate one of the variables by adding the equations together. This results in a single equation with one variable, which we can solve. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. Let's apply the elimination method to the same system:

-3x + 6y = 9
5x + 7y = -49

Step 1: Multiply the equations by constants so that the coefficients of one variable are opposites.

Let's eliminate x. To do this, we need to make the coefficients of x opposites. We can multiply the first equation by 5 and the second equation by 3:

5(-3x + 6y) = 5(9)  =>  -15x + 30y = 45
3(5x + 7y) = 3(-49) =>  15x + 21y = -147

Now, the coefficients of x are -15 and 15, which are opposites.

Step 2: Add the equations to eliminate one variable.

Add the two modified equations together:

(-15x + 30y) + (15x + 21y) = 45 + (-147)
51y = -102

Step 3: Solve the resulting equation for the remaining variable.

Solve for y:

y = -102 / 51
y = -2

Step 4: Substitute the value back into one of the original equations to find the other variable.

Now that we have y = -2, substitute this value back into the first original equation, -3x + 6y = 9:

-3x + 6(-2) = 9
-3x - 12 = 9
-3x = 21
x = -7

Thus, using the elimination method, we find the solution to the system of equations is x = -7 and y = -2. This confirms our result from the substitution method, highlighting the consistency of these techniques.

Verifying the Solution

To ensure accuracy, it's always good practice to verify the solution by substituting the values of x and y back into both original equations. If the values satisfy both equations, then the solution is correct. Let's verify our solution (-7, -2) for the system:

-3x + 6y = 9
5x + 7y = -49

Equation 1: -3x + 6y = 9

Substitute x = -7 and y = -2:

-3(-7) + 6(-2) = 9
21 - 12 = 9
9 = 9  (True)

Equation 2: 5x + 7y = -49

Substitute x = -7 and y = -2:

5(-7) + 7(-2) = -49
-35 - 14 = -49
-49 = -49  (True)

Since the values x = -7 and y = -2 satisfy both equations, we can confidently conclude that (-7, -2) is the correct solution to the system of equations.

Choosing the Right Method

Both the substitution and elimination methods are effective for solving systems of linear equations. The choice of method often depends on the specific system and personal preference. Here are some guidelines:

  • Substitution Method: This method is particularly useful when one of the equations is already solved for one variable or can be easily solved for one variable.
  • Elimination Method: This method is advantageous when the coefficients of one of the variables are either the same or opposites, or when it is easy to make them so by multiplying the equations by constants.

In the given system, both methods work well. However, some might find the elimination method slightly more straightforward due to the ease of making the coefficients of x opposites. Ultimately, the best method is the one you feel most comfortable and confident using.

Graphical Interpretation

As mentioned earlier, each linear equation in a system represents a straight line in the coordinate plane. The solution to the system corresponds to the point where the lines intersect. If we were to graph the equations -3x + 6y = 9 and 5x + 7y = -49, we would find that they intersect at the point (-7, -2). This graphical representation provides a visual confirmation of our algebraic solution.

Real-World Applications

Solving systems of equations is not just a mathematical exercise; it has numerous real-world applications. Here are a few examples:

  • Economics: Systems of equations are used to model supply and demand, determine equilibrium prices, and analyze market behavior.
  • Engineering: Engineers use systems of equations to design structures, analyze circuits, and model fluid flow.
  • Physics: Systems of equations are essential for solving problems in mechanics, electromagnetism, and thermodynamics.
  • Computer Science: Systems of equations are used in computer graphics, optimization algorithms, and machine learning.

By mastering the techniques for solving systems of equations, you gain a valuable skill that can be applied to a wide range of practical problems.

Conclusion

Solving systems of equations is a fundamental skill in mathematics with far-reaching applications. In this article, we explored the substitution and elimination methods, demonstrating how to solve a specific system of linear equations step-by-step. We also emphasized the importance of verifying the solution and discussed the graphical interpretation of systems of equations. By understanding these methods and their applications, you can confidently tackle a wide range of mathematical and real-world problems. The solution to the system

-3x + 6y = 9
5x + 7y = -49

is indeed (-7, -2), as we have shown using both the substitution and elimination methods. This comprehensive guide should serve as a valuable resource for anyone seeking to master the art of solving systems of equations. Remember, practice is key to proficiency, so work through various examples and challenge yourself with more complex systems. With dedication and the right techniques, you can become a confident problem solver in this essential area of mathematics.