Solving systems of equations with three variables is a powerful tool for tackling complex problems. By using methods like elimination, substitution, and Gaussian elimination, you can find solutions to interconnected equations.

These techniques are crucial for real-world applications, from economics to engineering. Understanding how to verify solutions and interpret results helps you apply these skills to practical situations, making math a valuable problem-solving tool.

Solving Systems of Equations with Three Variables

Verification of ordered triples

• An ordered triple $(x, y, z)$ represents a potential solution to a system of three linear equations
• Substitute the values of $x$, $y$, and $z$ from the ordered triple into each equation in the system
• Simplify the equations after substitution by performing arithmetic operations
• Check if the resulting equations are true statements that hold for the given values
  • If all three equations are satisfied, the ordered triple is a solution to the system ($x = 2$, $y = -1$, $z = 3$)
  • If any equation is not satisfied, the ordered triple is not a solution to the system ($x = 1$, $y = 0$, $z = 2$)

Methods for three-variable systems

• Elimination method eliminates one variable at a time by adding or subtracting equations (also known as simultaneous equations)
  • Choose two equations and multiply one or both by a constant to make the coefficients of one variable opposite
  • Add or subtract the equations to eliminate the variable with opposite coefficients
  • Repeat the process with another pair of equations to eliminate the same variable
  • Solve the resulting system of two equations with two variables using substitution or elimination
  • Substitute the values of the two variables into one of the original equations to find the value of the third variable
• Substitution method solves for one variable in terms of the others and substitutes the expression into the other equations
  1. Solve one equation for one of the variables in terms of the other two ($x = 2y + 3z$)
  2. Substitute the expression for the solved variable into the other two equations
  3. Solve the resulting system of two equations with two variables using substitution or elimination
  4. Substitute the values of the two variables into the expression for the third variable to find its value
• Gaussian elimination transforms the system into row echelon form using elementary row operations
  • Write the system of equations in augmented matrix form $[A|b]$
  • Use elementary row operations to transform the matrix into row echelon form
    • Swap rows to ensure the first non-zero entry in each row is 1 (pivot)
    • Multiply rows by non-zero constants to make the pivot entries 1
    • Add multiples of rows to other rows to eliminate entries above and below the pivots
  • Use back-substitution to find the values of the variables starting from the bottom row

Advanced Techniques in Linear Systems

• Matrix algebra provides a compact way to represent and manipulate systems of linear equations
• Linear algebra offers a framework for understanding and solving systems of equations in higher dimensions
• Determinants can be used to analyze the nature of solutions in a system of linear equations
• Cramer's rule provides an alternative method for solving systems of linear equations using determinants

Real-world applications of linear systems

• Identify the unknown quantities in the problem and assign variables to represent them ($x$: apples, $y$: bananas, $z$: oranges)
• Write a system of three linear equations based on the given information and relationships between the variables
  • Each equation should represent a distinct piece of information from the problem (total cost, total weight, total number of fruits)
• Solve the system of equations using elimination, substitution, or Gaussian elimination methods
• Interpret the solution in the context of the original problem
  • Verify that the solution makes sense and satisfies the given conditions (non-negative quantities, integer values if required)
  • If there is no solution, the problem may have conflicting information or be unsolvable (inconsistent system)
  • If there are infinitely many solutions, the problem may not provide enough information to determine a unique solution (dependent system)