Lesson 17.2

Multivariable Linear Systems

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

Introduction

In 3D space, we need three equations to pin down a single point. Gaussian elimination systematically reduces a system to triangular form, where the last equation has only one variable. From there, weback-substitute upward to find all unknowns—a technique that scales to systems of any size.

Prerequisite Connection

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

Today's Increment

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

Why This Matters

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 ().