Algebra 42 - Visualizing Linear Equations in Three Variables

TL;DR

This lecture introduces systems of three linear equations in three variables and explains how they can be graphically represented as planes in Cartesian coordinates.

Transcript

Hello. I'm Professor Von Schmohawk and welcome to Why U. In the last several lectures, we studied systems of two linear equations in two variables and saw how these systems could be represented graphically as lines in the Cartesian plane. Systems of equations can consist of more than two equations and the equations in the system can contain more th... Read More

Key Insights

• 👔 Linear equations in three variables are written in the form "Ax + By + Cz = D" where A, B, and C are coefficients.
• 👾 The graph of a linear equation in three variables is a plane in 3-dimensional Cartesian space.
• 😥 The solution set of a linear equation in three variables consists of all points in space that satisfy the equation.
• 😥 Systems of three linear equations in three variables can be solved graphically by finding the points where the corresponding planes intersect.
• 👾 Depending on the coefficients, the planes can be positioned and oriented differently in space.
• 😥 The points where all three planes simultaneously intersect correspond to the solutions of the system.
• 📈 The graph of a linear equation will be a plane as long as the coefficients A, B, and C are not all zero.

Questions & Answers

Q: What is the standard form of a linear equation in three variables?

The standard form of a linear equation in three variables is "Ax + By + Cz = D", where A, B, and C are coefficients and at least one of them must be non-zero.

Q: How are linear equations in three variables graphically represented?

Linear equations in three variables are represented as planes in 3-dimensional Cartesian space.

Q: What is the solution set of a linear equation in three variables?

The solution set of a linear equation in three variables is graphically represented as a plane. It consists of every point in 3-dimensional space that satisfies the equation.

Q: How are systems of three linear equations in three variables solved graphically?

Each equation in the system is represented as a plane, and the solutions are the points where all three planes intersect in 3-dimensional space.

Summary & Key Takeaways

• Systems of three linear equations in three variables can be represented as planes in 3-dimensional Cartesian space.

• The standard form of a linear equation in three variables is written as "Ax + By + Cz = D" where at least one of the coefficients, A, B, or C must be non-zero.

• When graphed, the solution set of a linear equation in three variables is a plane, and the points where three planes simultaneously intersect correspond to solutions of the system.