11.2 Systems of Linear Equations: Three Variables

Systems of linear equations with three variables expand our problem-solving toolkit. These systems involve three equations and three unknowns, allowing us to model more complex real-world scenarios. We'll explore methods like elimination and substitution to solve these systems.

Understanding the consistency of three-equation systems is crucial. We'll learn to distinguish between consistent systems with unique or infinite solutions, and inconsistent systems with no solutions. This knowledge helps us interpret results and apply them to practical situations.

Solving Systems of Linear Equations with Three Variables

Solving three-variable linear systems

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• Elimination method
  • Multiply equations by appropriate constants eliminates one variable at a time ($x$, $y$, or $z$)
  • Add or subtract resulting equations obtains an equation with two variables
  • Repeat process eliminates another variable, resulting in equation with one variable
  • Solve for remaining variable and substitute back finds values of other variables
• Substitution method
  • Solve one equation for one variable in terms of other two (express $x$ in terms of $y$ and $z$)
  • Substitute expression for solved variable into other two equations
  • Solve resulting system of two equations with two variables using substitution or elimination
  • Substitute values of two variables back into expression for third variable finds its value

Consistency of three-equation systems

• Consistent system
  • Has at least one solution (unique solution or infinitely many solutions)
  • Equations represent planes that intersect at a point (one solution) or a line (infinite solutions)
• Inconsistent system
  • Has no solution (equations have no common point of intersection)
  • Equations represent parallel planes that do not intersect
• Dependent system
  • Has infinite solutions (equations represent same plane or planes that intersect along a line)
  • Equations are multiples of each other or one equation can be derived from others

Interpreting solutions for three-equation systems

• One solution (consistent and independent)
  • Solution is an ordered triple $(x, y, z)$ satisfies all three equations simultaneously
  • Graphically, solution represents point of intersection of three planes
• No solution (inconsistent)
  • System has no solution, equations have no common point of intersection
  • Graphically, planes represented by equations are parallel and do not intersect
• Infinite solutions (consistent and dependent)
  • System has infinite number of solutions, represented by a line or a plane
  • Graphically, planes represented by equations coincide or intersect along a line
  • Express solution using parametric equations or as linear combination of variables ($x = a + bt, y = c + dt, z = e + ft$, where $t$ is a parameter)
  • One variable may be a free variable, allowing it to take any value while others are expressed in terms of it

Matrix Representation and Row Reduction

• Coefficient matrix: Organize coefficients of variables and constants into a matrix form
• Echelon form: Transform the coefficient matrix through row operations
  • Leading coefficients (pivots) are 1
  • Each leading 1 is in a column to the right of the leading 1 in the row above it
• Reduced row echelon form: Further simplification of echelon form
  • Each column containing a leading 1 has zeros in all other entries
  • Simplifies solving for variables and identifying free variables

Applying Systems of Linear Equations with Three Variables

Solving real-world problems with three-variable linear systems

1. Identify variables and quantities they represent in problem (let $x$ = number of apples, $y$ = number of bananas, $z$ = number of oranges)
2. Set up system of three linear equations based on given information and relationships between variables
3. Solve system using elimination, substitution, or other appropriate methods
4. Interpret solution in context of original problem, ensuring values make sense in given situation (negative numbers of fruit are not possible)