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 applications spanning various fields, from engineering and physics to economics and computer science. A system of equations is a set of two or more equations that share the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. This article will delve into the methods for solving systems of equations, with a detailed walkthrough of solving the specific system provided, and offer insights to help you master this essential mathematical concept.

Understanding Systems of Equations

Before diving into specific methods, it's crucial to understand what a system of equations represents. Each equation in the system defines a relationship between the variables. In the case of two variables, each equation can be visualized as a line on a coordinate plane. The solution to the system, if it exists, corresponds to the point(s) where the lines intersect. There are three possible scenarios when dealing with a system of two linear equations:

  1. Unique Solution: The lines intersect at a single point, indicating a unique solution.
  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), meaning every point on the line is a solution.

When approaching a system of equations, the goal is to find the values of the variables that make all equations true. This can be achieved through various algebraic methods, which we will explore in the following sections.

Methods for Solving Systems of Equations

Several methods exist for solving systems of equations, each with its own advantages and suitability depending on the specific system. The most common methods include:

  1. Substitution Method: This method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily solved. The value obtained is then substituted back into one of the original equations to find the value of the other variable.

  2. Elimination Method (also known as the Addition Method): This method involves manipulating the equations so that the coefficients of one variable are opposites. By adding the equations together, one variable is eliminated, leaving a single equation with one variable. This equation can be solved, and the value obtained is then substituted back into one of the original equations to find the value of the other variable.

  3. Graphing Method: This method involves graphing each equation on a coordinate plane. The solution to the system is the point(s) where the graphs intersect. While visually intuitive, this method is most accurate for linear equations and may not provide precise solutions for non-linear systems or when dealing with fractional or irrational solutions.

  4. Matrix Methods: For larger systems of equations, matrix methods such as Gaussian elimination or matrix inversion can be more efficient. These methods involve representing the system of equations in matrix form and using matrix operations to solve for the variables. These methods are particularly useful when dealing with systems of three or more equations.

In the following sections, we will focus on the substitution and elimination methods as they are particularly effective for solving the given system of equations.

Solving the System: Elimination Method

Let's apply the elimination method to solve the given system of equations:

5x - 2y = 18
3x + 3y = 15

The elimination method aims to eliminate one variable by manipulating the equations so that the coefficients of one variable are opposites. Observe that if we multiply the first equation by 3 and the second equation by 2, the coefficients of y will become -6 and 6, respectively.

Multiply the first equation by 3:

3(5x - 2y) = 3(18)
15x - 6y = 54

Multiply the second equation by 2:

2(3x + 3y) = 2(15)
6x + 6y = 30

Now we have the following system:

15x - 6y = 54
6x + 6y = 30

Adding the two equations together, we eliminate y:

(15x - 6y) + (6x + 6y) = 54 + 30
21x = 84

Divide both sides by 21 to solve for x:

x = 84 / 21
x = 4

Now that we have found the value of x, we can substitute it back into either of the original equations to solve for y. Let's use the second equation:

3x + 3y = 15
3(4) + 3y = 15
12 + 3y = 15

Subtract 12 from both sides:

3y = 15 - 12
3y = 3

Divide both sides by 3 to solve for y:

y = 3 / 3
y = 1

Therefore, the solution to the system of equations is x = 4 and y = 1, which can be written as the ordered pair (4, 1).

Solving the System: Substitution Method

Alternatively, we can solve the same system of equations using the substitution method. This method involves solving one equation for one variable and substituting that expression into the other equation.

Starting with the system:

5x - 2y = 18
3x + 3y = 15

Let's solve the second equation for y:

3x + 3y = 15
3y = 15 - 3x
y = (15 - 3x) / 3
y = 5 - x

Now substitute this expression for y into the first equation:

5x - 2y = 18
5x - 2(5 - x) = 18

Distribute the -2:

5x - 10 + 2x = 18

Combine like terms:

7x - 10 = 18

Add 10 to both sides:

7x = 28

Divide both sides by 7 to solve for x:

x = 28 / 7
x = 4

Now substitute the value of x back into the expression we found for y:

y = 5 - x
y = 5 - 4
y = 1

Again, we find that the solution to the system of equations is x = 4 and y = 1, or the ordered pair (4, 1).

Verifying the Solution

It's always a good practice to verify the solution by substituting the values of x and y back into both original equations to ensure they are satisfied.

For the first equation:

5x - 2y = 18
5(4) - 2(1) = 18
20 - 2 = 18
18 = 18  (True)

For the second equation:

3x + 3y = 15
3(4) + 3(1) = 15
12 + 3 = 15
15 = 15  (True)

Since the solution (4, 1) satisfies both equations, we can confidently conclude that it is the correct solution to the system.

Conclusion

In this article, we explored the process of solving systems of equations, focusing on the elimination and substitution methods. We demonstrated how to apply these methods to solve the specific system:

5x - 2y = 18
3x + 3y = 15

We found that the solution is (4, 1). Understanding these methods is essential for success in algebra and beyond. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical problems and real-world applications involving systems of equations. Remember to always verify your solution to ensure accuracy. Practice is key to mastering these skills, so work through various examples to solidify your understanding. The ability to solve systems of equations is a valuable asset in any field that requires analytical thinking and problem-solving skills. Whether you're a student, engineer, scientist, or simply someone who enjoys mathematical challenges, the knowledge and techniques discussed in this article will serve you well.