Systems of linear equations with three variables expand on two-variable systems. They involve solving for x, y, and z simultaneously using methods like elimination, substitution, or linear combination. These systems can have unique, infinite, or no solutions.

Analyzing three-variable systems requires understanding consistency and dependency. Consistent systems have solutions, while inconsistent ones don't. Dependent systems have infinitely many solutions, often represented using parametric equations. Visualizing solutions in 3D space helps interpret the results.

Solving Systems of Linear Equations with Three Variables

Solving three-variable linear systems

• Elimination method involves multiplying equations by constants to eliminate one variable at a time
  • Add or subtract the resulting equations to obtain an equation with two variables
  • Repeat the process to eliminate another variable, resulting in an equation with one variable
  • Solve for the remaining variable and substitute back to find the values of the other variables (e.g., $x = 2$, $y = -1$, $z = 3$)
• Substitution method involves solving one equation for one of the variables in terms of the other two
  • Substitute the expression for the solved variable into the other two equations
  • Solve the resulting system of two equations with two variables using substitution or elimination
  • Substitute the values of the two variables back into the expression for the third variable to find its value (e.g., $x = 5$, $y = 3$, $z = -2$)
• Linear combination method involves expressing the solution as a combination of the original equations

Consistency of three-equation systems

• Consistent system has at least one solution
  • Equations represent planes that intersect at a point (one solution) or a line (infinitely many solutions)
  • Example: $x + y + z = 6$, $2x - y + z = 5$, $x - 2y + 3z = 7$ has a unique solution $(1, 2, 3)$
• Inconsistent system has no solution
  • Equations represent planes that do not intersect
  • Attempting to solve the system results in a contradiction, such as $0 = 1$
  • Example: $x + y + z = 4$, $2x + 2y + 2z = 9$, $3x + 3y + 3z = 12$ has no solution
• Dependent system has infinitely many solutions
  • Equations represent planes that coincide or overlap
  • One equation can be derived from the others by multiplying by a constant or adding/subtracting the other equations
  • Example: $x + y + z = 5$, $2x + 2y + 2z = 10$, $3x + 3y + 3z = 15$ has infinitely many solutions

Interpreting Solutions for Dependent Systems

Solutions for dependent linear systems

• Dependent systems have infinitely many solutions forming a line or a plane in three-dimensional space
• Express solutions using parametric equations by choosing one or two variables as parameters ($t$ or $s$ and $t$)
  • Express the other variables in terms of the chosen parameter(s)
  • Example: If $x = 2t$, $y = 3t$, and $z = t$, the solution can be written as $(x, y, z) = (2t, 3t, t)$ for any real value of $t$
• Interpret the solution set as a line or a plane in three-dimensional space
  • Each point on the line or plane corresponds to a specific value of the parameter(s) and satisfies all three original equations simultaneously
  • Example: The system $x + y + z = 6$, $2x + 2y + 2z = 12$, $3x + 3y + 3z = 18$ has infinitely many solutions of the form $(x, y, 6-x-y)$ for any real values of $x$ and $y$
• The geometric interpretation of the solution set can be visualized as the intersection of planes in 3D space

Matrix Representation and Analysis

• The coefficient matrix of a system contains the coefficients of the variables in each equation
• Reduced row echelon form simplifies the coefficient matrix to analyze the system's solutions
• The rank of a matrix determines the number of linearly independent equations in the system
• A homogeneous system has the constant term equal to zero in all equations