In this section, we will explore systems of linear equations, which are collections of linear equations involving the same set of variables. Understanding how to solve these systems is crucial for many applications in 3D mathematics, including computer graphics, physics simulations, and more.

A system of linear equations consists of two or more linear equations with the same set of variables. For example:

\[ \begin{cases} 2x + 3y - z = 5
4x - y + 2z = 6
-x + 2y + 3z = 4 \end{cases} \]

A solution to a system of linear equations is a set of values for the variables that satisfies all the equations simultaneously. There are three possible types of solutions:

1. Unique Solution: There is exactly one set of values that satisfies all equations.
2. No Solution: There is no set of values that satisfies all equations.
3. Infinite Solutions: There are infinitely many sets of values that satisfy all equations.

1. Graphical Method: Plotting each equation on a graph to find the point(s) of intersection.
2. Substitution Method: Solving one equation for one variable and substituting this value into the other equations.
3. Elimination Method: Adding or subtracting equations to eliminate one variable, making it easier to solve for the others.
4. Matrix Method: Using matrices and operations like row reduction to solve the system.

The matrix method is particularly useful for solving larger systems of equations. It involves representing the system as a matrix equation and then using techniques such as Gaussian elimination to find the solution.

Consider the system:

\[ \begin{cases} 2x + 3y - z = 5
4x - y + 2z = 6
-x + 2y + 3z = 4 \end{cases} \]

This can be written in matrix form as:

\[ \begin{bmatrix} 2 & 3 & -1
4 & -1 & 2
-1 & 2 & 3 \end{bmatrix} \begin{bmatrix} x
y
z \end{bmatrix}

\begin{bmatrix} 5
6
4 \end{bmatrix} \]

Or more compactly as:

\[ A\mathbf{x} = \mathbf{b} \]

where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the vector of constants.

Gaussian elimination is a method for solving the matrix equation \( A\mathbf{x} = \mathbf{b} \). It involves three main steps:

1. Form the augmented matrix: Combine the coefficient matrix \( A \) and the constants vector \( \mathbf{b} \) into an augmented matrix \([A|\mathbf{b}]\).

2. Row reduction: Use row operations to transform the augmented matrix into row-echelon form.

3. Back substitution: Solve for the variables starting from the last row and working upwards.

Example

Solve the system:

\[ \begin{cases} 2x + 3y - z = 5
4x - y + 2z = 6
-x + 2y + 3z = 4 \end{cases} \]

Step 1: Form the augmented matrix

\[ \left[\begin{array}{ccc|c} 2 & 3 & -1 & 5
4 & -1 & 2 & 6
-1 & 2 & 3 & 4 \end{array}\right] \]

Step 2: Row reduction

• Multiply the first row by 2 and subtract from the second row:

\[ \left[\begin{array}{ccc|c} 2 & 3 & -1 & 5
0 & -7 & 4 & -4
-1 & 2 & 3 & 4 \end{array}\right] \]

• Multiply the first row by -1 and add to the third row:

\[ \left[\begin{array}{ccc|c} 2 & 3 & -1 & 5
0 & -7 & 4 & -4
0 & 5 & 2 & 9 \end{array}\right] \]

• Multiply the second row by -1/7:

\[ \left[\begin{array}{ccc|c} 2 & 3 & -1 & 5
0 & 1 & -4/7 & 4/7
0 & 5 & 2 & 9 \end{array}\right] \]

• Multiply the second row by 5 and subtract from the third row:

\[ \left[\begin{array}{ccc|c} 2 & 3 & -1 & 5
0 & 1 & -4/7 & 4/7
0 & 0 & 34/7 & 53/7 \end{array}\right] \]

• Multiply the third row by 7/34:

\[ \left[\begin{array}{ccc|c} 2 & 3 & -1 & 5
0 & 1 & -4/7 & 4/7
0 & 0 & 1 & 53/34 \end{array}\right] \]

Step 3: Back substitution

• From the third row: \( z = 53/34 \)
• Substitute \( z \) into the second row: \( y = 4/7 + 4/7 \cdot 53/34 = 2 \)
• Substitute \( y \) and \( z \) into the first row: \( x = (5 + 1 - 3 \cdot 2)/2 = 0 \)

Thus, the solution is:

\[ \begin{bmatrix} x
y
z \end{bmatrix}

\begin{bmatrix} 0
2
53/34 \end{bmatrix} \]

Solve the following system of linear equations using Gaussian elimination:

\[ \begin{cases} x + 2y + 3z = 9
2x + 3y + z = 8
3x + y + 2z = 7 \end{cases} \]

Solution:

1. Form the augmented matrix:

\[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 9
2 & 3 & 1 & 8
3 & 1 & 2 & 7 \end{array}\right] \]

2. Perform row operations to reach row-echelon form and solve using back substitution.

Determine if the following system has a unique solution, no solution, or infinitely many solutions:

\[ \begin{cases} x + y + z = 3
2x + 2y + 2z = 6
x - y + z = 1 \end{cases} \]

Solution:

1. Form the augmented matrix:

\[ \left[\begin{array}{ccc|c} 1 & 1 & 1 & 3
2 & 2 & 2 & 6
1 & -1 & 1 & 1 \end{array}\right] \]

2. Perform row operations to determine the nature of the solution.

In this section, we have covered the fundamentals of systems of linear equations, including their definition, types of solutions, and various methods for solving them. We focused on the matrix method and Gaussian elimination, which are powerful tools for solving larger systems. Understanding these concepts is essential for further studies in 3D mathematics and their applications in computer graphics and other fields.