Solving Systems Of Equations A Step-by-Step Guide

Jul 6, 2025 by Admin 50 views

Solving the System of Equations: A Comprehensive 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. This article delves into a comprehensive approach to solving the given system of equations, providing a step-by-step guide and explanations to enhance understanding. We will explore the methods used to find the values of the variables that satisfy all equations simultaneously.

Understanding Systems of Equations

Before diving into the solution, let's establish a clear understanding of what a system of equations entails. A system of equations is a set of two or more equations containing the same variables. The goal is to find the values for these variables that make all equations true. These values, when found, represent the point(s) where the lines or planes represented by the equations intersect.

The system we are tasked with solving is:

-3x + 3y = -5/8
x - 4z = -5/3
-4y + z = 0
(x, y, z) = 1

This system consists of three linear equations with three unknowns (x, y, and z), along with the condition (x, y, z) = 1, which we will interpret as a constraint that the greatest common divisor (GCD) of the solutions should be 1, indicating we are looking for integer solutions in their simplest form. Solving such a system requires a systematic approach to eliminate variables and arrive at a unique solution.

Methods for Solving Systems of Equations

Several methods exist for solving systems of equations, including substitution, elimination, and matrix methods. Each method has its advantages and is suitable for different types of systems. For this particular system, we will primarily use the substitution and elimination methods, which are well-suited for linear equations.

Substitution Method: This method involves solving one equation for one variable and substituting that expression into another equation. This reduces the number of variables in the second equation, making it easier to solve. We repeat this process until we have a single equation with one variable.
Elimination Method: This method involves adding or subtracting multiples of the equations to eliminate one of the variables. This results in a new equation with fewer variables, simplifying the system. We continue this process until we can solve for the remaining variables.
Matrix Methods: These methods involve representing the system of equations in matrix form and using techniques like Gaussian elimination or finding the inverse of the matrix to solve for the variables. These methods are particularly useful for larger systems of equations.

Step-by-Step Solution

Step 1: Simplify the Equations

The first step is to simplify the equations to make them easier to work with. We can start by eliminating the fractions in the first two equations. Multiply the first equation by 8 and the second equation by 3:

8 * (-3x + 3y) = 8 * (-5/8)
3 * (x - 4z) = 3 * (-5/3)

This simplifies to:

-24x + 24y = -5   (Equation 1)
3x - 12z = -5     (Equation 2)
-4y + z = 0      (Equation 3)

Step 2: Use Substitution or Elimination

We can use the substitution or elimination method to reduce the system to two equations with two variables. From Equation 3, we can express z in terms of y:

z = 4y         (Equation 4)

Now, substitute Equation 4 into Equation 2:

3x - 12(4y) = -5
3x - 48y = -5     (Equation 5)

Step 3: Solve for x and y

Now we have two equations with two variables (x and y):

-24x + 24y = -5   (Equation 1)
3x - 48y = -5     (Equation 5)

To eliminate x, multiply Equation 5 by 8:

8 * (3x - 48y) = 8 * (-5)
24x - 384y = -40   (Equation 6)

Add Equation 1 and Equation 6:

(-24x + 24y) + (24x - 384y) = -5 + (-40)
-360y = -45
y = -45 / -360
y = 1/8

Now that we have the value of y, we can substitute it back into Equation 5 to find x:

3x - 48(1/8) = -5
3x - 6 = -5
3x = 1
x = 1/3

Step 4: Solve for z

Substitute the value of y into Equation 4 to find z:

z = 4y
z = 4(1/8)
z = 1/2

Step 5: Check the Solutions

Now we have the values x = 1/3, y = 1/8, and z = 1/2. Let's check if these values satisfy the original equations:

Equation 1: -3x + 3y = -5/8

-3(1/3) + 3(1/8) = -1 + 3/8 = -8/8 + 3/8 = -5/8 (Correct)

Equation 2: x - 4z = -5/3

(1/3) - 4(1/2) = 1/3 - 2 = 1/3 - 6/3 = -5/3 (Correct)

Equation 3: -4y + z = 0

-4(1/8) + (1/2) = -1/2 + 1/2 = 0 (Correct)

The solutions satisfy all three equations.

Step 6: Verify the GCD Condition

Finally, we need to verify that the greatest common divisor (GCD) of the solutions is 1. The solutions are x = 1/3, y = 1/8, and z = 1/2. To find the GCD, we first express the solutions as integers by finding a common denominator. The least common multiple of 3, 8, and 2 is 24. So, we multiply each fraction by 24:

x = (1/3) * 24 = 8
y = (1/8) * 24 = 3
z = (1/2) * 24 = 12

Now we find the GCD of 8, 3, and 12. The GCD(8, 3, 12) = 1. Thus, the condition is satisfied.

Alternative Method: Matrix Representation

For a more streamlined approach, especially with larger systems, we can represent the system in matrix form and use matrix operations to solve it. The given system can be written in matrix form as:

| -3  3  0 | | x | = | -5/8 |
|  1  0 -4 | | y | = | -5/3 |
|  0 -4  1 | | z | = |  0   |

This can be written as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix:

A = | -3  3  0 |
    |  1  0 -4 |
    |  0 -4  1 |

X = | x |
    | y |
    | z |

B = | -5/8 |
    | -5/3 |
    |  0   |

To solve for X, we can use the formula X = A^(-1)B, where A^(-1) is the inverse of matrix A. First, we need to find the determinant of A:

det(A) = -3(0*1 - (-4)*(-4)) - 3(1*1 - (-4)*0) + 0 = -3(-16) - 3(1) = 48 - 3 = 45

Since the determinant is non-zero, the inverse exists. The inverse of A can be calculated as:

A^(-1) = (1/det(A)) * adj(A)

Where adj(A) is the adjugate of A, which is the transpose of the cofactor matrix of A. The cofactor matrix is:

C = | -16 -1 -4 |
    | -3  -3 -12 |
    | -12 -12 -3 |

The adjugate of A is the transpose of C:

adj(A) = | -16 -3 -12 |
         | -1  -3 -12 |
         | -4 -12 -3 |

So, the inverse of A is:

A^(-1) = (1/45) * | -16 -3 -12 |
               | -1  -3 -12 |
               | -4 -12 -3 |

Now, we can find X by multiplying A^(-1) with B:

X = A^(-1)B = (1/45) * | -16 -3 -12 | * | -5/8 |
                  | -1  -3 -12 |   | -5/3 |
                  | -4 -12 -3 |   |  0   |

X = (1/45) * | (-16)*(-5/8) + (-3)*(-5/3) + (-12)*(0) |
                  | (-1)*(-5/8) + (-3)*(-5/3) + (-12)*(0) |
                  | (-4)*(-5/8) + (-12)*(-5/3) + (-3)*(0) |

X = (1/45) * | 10 + 5 + 0 |
                  | 5/8 + 5 + 0 |
                  | 5/2 + 20 + 0 |

X = (1/45) * | 15 |
                  | 45/8 |
                  | 45/2 |

X = | 15/45 |
    | (45/8)/45 |
    | (45/2)/45 |

X = | 1/3 |
    | 1/8 |
    | 1/2 |

Thus, x = 1/3, y = 1/8, and z = 1/2, which matches our previous solution.

Common Pitfalls and How to Avoid Them

Solving systems of equations can be tricky, and there are several common pitfalls to watch out for:

Arithmetic Errors: Simple arithmetic errors can lead to incorrect solutions. Always double-check your calculations, especially when dealing with fractions or negative numbers.
Incorrect Substitution: Ensure you are substituting correctly and into the right equations. A mistake in substitution can throw off the entire solution.
Forgetting to Check Solutions: Always check your solutions by plugging them back into the original equations. This helps catch any errors made during the solving process.
Misinterpreting the GCD Condition: The condition (x, y, z) = 1 means the GCD of the integer forms of x, y, and z should be 1. Make sure to convert your solutions to integers before checking this condition.

Conclusion

Solving systems of equations is a vital skill in mathematics and various applied fields. By understanding the different methods available, such as substitution, elimination, and matrix methods, and by following a systematic approach, one can effectively solve complex systems. In this article, we provided a detailed step-by-step solution to the given system, along with explanations and checks to ensure accuracy. The solutions x = 1/3, y = 1/8, and z = 1/2 satisfy all the equations and the GCD condition, demonstrating a comprehensive understanding of the problem and its solution.

By mastering the techniques discussed in this guide, readers can confidently tackle a wide range of systems of equations, enhancing their problem-solving abilities and mathematical proficiency.