Introduction to Systems of Linear Equations

TL;DR

A system of linear equations is just a set of two or more equations that you want to solve at the same time. The goal is to find values for all the variables that work for every equation in the system. When we talk about 3x3 systems, we mean three equations with three different variables.

1. The Mental Model

Think of each equation as a rule, and a system of equations as a collection of rules that all have to be followed at once. You're trying to find a specific solution (like a secret code) that satisfies every single rule.

2. The Core Material

When you're dealing with a system of linear equations, you're looking for the point (or points) where all the lines (or planes, in 3D) intersect. For a 3x3 system, you're often looking for a single point in 3D space where three planes cross.

What is a Linear Equation?

Photo by Vanessa Garcia on Pexels

A linear equation is an equation where the highest power of any variable is 1. You won't see things like $x^2$, $\sqrt{y}$, or $1/z$. For example, $2x + 3y - z = 7$ is a linear equation. The graph of a linear equation with two variables is a straight line, and with three variables, it's a flat plane.

What Does a "Solution" Mean?

Photo by Ann H on Pexels

A solution to a system of equations is a set of values for all the variables that makes every equation in the system true. If you plug the solution values back into any equation in the system, that equation should hold.

Types of Solutions

Photo by Markus Winkler on Pexels

For a system of linear equations, there are generally three possibilities for the number of solutions:

graph TD
    Start["System of Linear Equations"] --> A["How many solutions?"]

    A --> B{"One Solution"}
    B --> B1["Lines/Planes intersect at a single point"]

    A --> C{"No Solution"}
    C --> C1["Lines/Planes are parallel and never intersect"]

    A --> D{"Infinitely Many Solutions"}
    D --> D1["Lines/Planes are the same or perfectly overlap"]

• One Solution: This is the most common case. All the lines or planes intersect at exactly one specific point. This means there's one unique set of values for your variables.
• No Solution: Sometimes, the lines or planes are parallel and never cross, or they cross in ways that don't satisfy all equations simultaneously. In this situation, there's no set of values that can make all equations true.
• Infinitely Many Solutions: This happens when the equations essentially describe the same line or plane, or they all intersect along a common line or plane. Any point on that common intersection is a solution.

What "3x3" Means

Photo by William Warby on Pexels

"3x3 system" simply means you have:
* Three linear equations.
* Three different variables (commonly $x$, $y$, and $z$).

Example of a 3x3 system:
$x + y + z = 6$
$2x - y + 3z = 9$
$-x + 2y - z = 2$

Your goal is to find the values of $x$, $y$, and $z$ that work for all three equations simultaneously.

3. Worked Example

Let's look at a simple example to see what a solution looks like. We won't solve it yet, but just check if a given set of values is a solution.

System of Equations:
1. $x + y + z = 6$
2. $2x - y + 3z = 9$
3. $-x + 2y - z = 2$

Is $(x=1, y=2, z=3)$ a solution to this system?

Let's plug these values into each equation:

Equation 1:
$1 + 2 + 3 = 6$
$6 = 6$ (True!)

Equation 2:
$2(1) - 2 + 3(3) = 9$
$2 - 2 + 9 = 9$
$0 + 9 = 9$
$9 = 9$ (True!)

Equation 3:
$-(1) + 2(2) - 3 = 2$
$-1 + 4 - 3 = 2$
$3 - 3 = 2$
$0 = 2$ (False!)

Since $(1, 2, 3)$ does not satisfy the third equation, it is not a solution to the system. A solution must work for all equations.

4. Key Takeaways

• A system of linear equations is a collection of equations you solve simultaneously.
• A 3x3 system has three linear equations and three variables.
• A solution must make every equation in the system true.
• Geometrically, you're finding where all lines or planes intersect.
• Systems can have one solution, no solutions, or infinitely many solutions.
• If even one equation isn't satisfied by a set of values, those values aren't the solution.

5. Now Try It

Consider the system:
1. $x - y + z = 2$
2. $3x + y - 2z = 9$
3. $2x - 2y + z = 3$

Without solving, determine if the point $(x=3, y=2, z=1)$ is a solution to this system. Write down your steps for checking each equation.

Success looks like: Clearly showing your substitutions into each of the three equations and stating whether the given point is or is not a solution, with a reason.