Solving System Of Equations - A Comprehensive Guide

JU07/21/2025 08, 2025 by THE IDEN 52 views

Solving Systems of Equations: A Step-by-Step Guide

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. In this comprehensive guide, we will delve into the intricacies of solving systems of equations, focusing on a specific example to illustrate the process. Our aim is to equip you with the knowledge and confidence to tackle similar problems effectively.

Understanding Systems of Equations

Before we dive into the solution, let's first grasp the essence of systems of equations. A system of equations is essentially a collection of two or more equations that involve the same set of variables. These equations represent relationships between the variables, and our goal is to find the values of these variables that make all the equations true at the same time. This concept is crucial in various real-world applications, where multiple variables are interconnected, and we need to find a set of values that satisfies all the conditions.

The number of equations and variables in a system can vary. A system can have two equations and two variables, three equations and three variables, or even more. The complexity of solving a system generally increases with the number of equations and variables. However, the underlying principle remains the same: we aim to find the values that satisfy all the equations concurrently.

Methods for Solving Systems of Equations

There are several methods available for solving systems of equations, each with its own strengths and weaknesses. The most common methods include:

Substitution: This method involves solving one equation for one variable and then substituting that expression into another equation. This reduces the system to a single equation with one variable, which can then be solved directly. The value obtained is then substituted back into one of the original equations to find the value of the other variable.
Elimination: The elimination method involves manipulating the equations in the system so that when they are added or subtracted, one of the variables is eliminated. This again reduces the system to a single equation with one variable. This method is particularly useful when the coefficients of one of the variables are the same or opposites in two equations.
Matrix Methods: For larger systems of equations, matrix methods like Gaussian elimination or matrix inversion are often employed. These methods provide a systematic way to solve the system using matrix operations.

The choice of method depends on the specific system of equations and the preferences of the solver. Some systems are more easily solved using substitution, while others are better suited for elimination. Matrix methods are generally used for larger systems where manual calculations become cumbersome.

The System at Hand

Now, let's turn our attention to the specific system of equations we will be solving:

-3x - y = 7
y - 2z = 8
x - 4y + z = -14

This system consists of three linear equations with three variables: x, y, and z. Our goal is to find the values of x, y, and z that satisfy all three equations simultaneously. This system is a classic example of a 3x3 linear system, which is commonly encountered in various mathematical and scientific contexts.

Step-by-Step Solution

We will use a combination of substitution and elimination methods to solve this system. Let's break down the solution process into manageable steps:

Step 1: Solve for a Variable in One Equation

Looking at the equations, we can easily solve the second equation for y:

y = 2z + 8

This step isolates y in terms of z, which will be useful for substitution later on. Choosing the right equation and variable to isolate can simplify the process significantly. In this case, the second equation was a good choice because it had a simple coefficient for y.

Step 2: Substitute into Other Equations

Now, we substitute the expression for y (2z + 8) into the first and third equations:

First equation: -3x - (2z + 8) = 7 => -3x - 2z - 8 = 7
Third equation: x - 4(2z + 8) + z = -14 => x - 8z - 32 + z = -14

This substitution eliminates y from the first and third equations, resulting in a system of two equations with two variables, x and z. The substitution step is a key technique in solving systems of equations, as it reduces the complexity of the system by eliminating variables.

Step 3: Simplify the Equations

Let's simplify the equations obtained in the previous step:

-3x - 2z - 8 = 7 => -3x - 2z = 15
x - 8z - 32 + z = -14 => x - 7z = 18

These simplified equations are easier to work with. Simplifying equations by combining like terms and moving constants to one side is a crucial step in solving systems of equations. It reduces the chances of making errors in subsequent steps.

Step 4: Eliminate Another Variable

Now, we can eliminate x from the two simplified equations. To do this, we multiply the second equation by 3:

3(x - 7z) = 3(18) => 3x - 21z = 54

Then, we add this modified equation to the first equation:

(-3x - 2z) + (3x - 21z) = 15 + 54
-23z = 69

This eliminates x, leaving us with a single equation in z. The elimination step is a powerful technique for solving systems of equations. By strategically manipulating the equations, we can eliminate variables and reduce the system to a simpler form.

Step 5: Solve for the Remaining Variable

Now, we solve for z:

-23z = 69 => z = -3

We have found the value of z! Solving for the remaining variable is the final step in reducing the system to a single value. Once we have the value of one variable, we can use it to find the values of the other variables.

Step 6: Back-Substitute to Find Other Variables

We can now back-substitute the value of z into the equation y = 2z + 8 to find y:

y = 2(-3) + 8 => y = 2

And finally, we substitute the values of y and z into the equation x - 4y + z = -14 to find x:

x - 4(2) + (-3) = -14 => x - 8 - 3 = -14 => x = -3

Back-substitution is a crucial step in solving systems of equations. Once we have the value of one variable, we can substitute it back into the previous equations to find the values of the other variables. This process allows us to unravel the relationships between the variables and find the complete solution to the system.

Step 7: Verify the Solution

It's always a good practice to verify the solution by substituting the values of x, y, and z back into the original equations:

-3(-3) - 2 = 7 => 9 - 2 = 7 (True)
2 - 2(-3) = 8 => 2 + 6 = 8 (True)
-3 - 4(2) + (-3) = -14 => -3 - 8 - 3 = -14 (True)

All three equations are satisfied, so our solution is correct.

The Solution

The solution to the system of equations is:

x = -3, y = 2, z = -3

This set of values satisfies all three equations in the system, making it the unique solution. The solution represents the point where the three planes represented by the equations intersect in three-dimensional space.

Conclusion

Solving systems of equations is a fundamental skill in mathematics with wide-ranging applications. By mastering the techniques of substitution and elimination, you can effectively solve a variety of systems, from simple 2x2 systems to more complex 3x3 systems and beyond. The key is to understand the underlying principles and to practice applying the methods systematically. Remember to always verify your solution to ensure accuracy.

In this guide, we have provided a step-by-step solution to a specific system of equations, illustrating the process in detail. By following this guide and practicing with other examples, you can develop your skills in solving systems of equations and confidently tackle a wide range of mathematical problems.

Whether you are a student learning algebra, a professional working in a technical field, or simply someone who enjoys problem-solving, the ability to solve systems of equations is a valuable asset. So, keep practicing, keep learning, and keep exploring the fascinating world of mathematics!