Lesson 17.2

Multivariable Linear Systems

Extending our methods to three or more variables using systematic elimination.

Introduction

Extending our methods to three or more variables using systematic elimination.

Past Knowledge

You can solve 2×2 systems using substitution and elimination.

Today's Goal

We extend to 3 variables using Gaussian elimination and back-substitution.

Future Success

3D geometry, circuit analysis, and multivariable calculus all require systems with 3+ unknowns.

Key Concepts

Triangular (Row Echelon) Form

Each equation has one fewer variable than the one above.

Back-Substitution Steps

Solve the LAST equation for its variable (easiest: )

Substitute into the second-to-last equation:

Continue upward until all variables are found

Worked Examples

Example 1: Back-Substitution (Basic)

The system is already in triangular form:

Step 1: From equation 3

Step 2: Substitute into equation 2

Step 3: Substitute into equation 1

Solution:

Example 2: Full Gaussian Elimination (Intermediate)

Solve:

Step 1: Eliminate from equations 2 and 3

Eq2 - 2×Eq1:

Eq3 - Eq1:

Step 2: Eliminate

Step 3: Back-substitute

Solution:

Example 3: Dependent System (Advanced)

Solve:

Observation

Equation 2 is just 2× Equation 1 → only 2 independent equations for 3 unknowns

Result

Infinitely many solutions along a line in 3D space. Express solutions in terms of a parameter .

Parametric Solution: for any

Common Pitfalls

Arithmetic errors during elimination

With 3 variables, there are many operations—double-check each step carefully.

Wrong back-substitution order

Always start from the LAST equation (with fewest variables) and work upward.

Missing dependent systems

If you get , check for infinitely many solutions (parametric form).

Real-World Application

Electrical Circuit Analysis (Kirchhoff's Laws)

Engineers use systems of 3+ equations to find currents in complex circuits. Each loop and junction generates one equation, and solving the system gives the current through each component.

Example: A circuit with 3 loops produces 3 equations in 3 unknowns ().