Systems of Linear Equations and Matrices

References:

Book – Elementary Linear Algebra by Howard Anton and Chris Rorres, Wiley

Linear Equation:

At its simplest, a linear equation is an algebraic equation that creates a straight line when plotted on a graph. Every variable in the equation is raised to the first power (meaning no exponents like $x^2$ or $y^3$), and there are no variables multiplied by each other.

The Standard Forms

Depending on how you are using it, a linear equation usually appears in one of two ways:

1. Slope-Intercept Form

This is the most common version used for graphing:

$y=mx+b$

$x$ and $y$: The coordinates of any point on the line.
$m$: The slope (how steep the line is).
$b$: The y-intercept (where the line crosses the vertical y-axis).

2. Standard Form

This version is often used in systems of equations:

$Ax+By=C$

Where $A$, $B$, and $C$ are constants.

Visualizing the Equation

If you change the values in the equation, the line moves.

Changing $m$: Adjusts the tilt. A positive m goes up (left to right), while a negative m goes down.
Changing $b$: Slides the line up or down the graph.

Interactive: $y = mx + b$

Slope ($m$): 1

Y-Intercept ($b$): 0

Key Characteristics

To spot a linear equation in the wild, look for these traits:

Constant Rate of Change: For every unit you move across ($x$), the value of y changes by the same amount.
No Curves: Because the variables don't have exponents higher than 1, the "path" never bends.
Single or Multiple Variables: It can be simple ($x=5$) or involve multiple dimensions ($3x+2y−z=10$), but it will always represent a "flat" shape (a line, a plane, etc.).

📈 Constant Rate

For every unit you move across ($x$), the value of $y$ changes by the same consistent amount. This is the "slope."

📏 No Curves

Variables never have exponents higher than 1 (no $x^2$ or $y^3$). Because of this, the "path" never bends.

🔳 "Flat" Geometry

Whether it's 2D ($x, y$) or 3D ($x, y, z$), it always represents a flat shape - like a straight line or a flat sheet.

Real-World Example

Imagine a taxi service that charges a flat fee of $5 just to get in the car, plus $2 for every mile driven. This is a linear equation:

$y=2x+5$

If you drive 0 miles, you pay $5.
If you drive 10 miles, you pay $25 (2×10+5).

🚕

The Taxi Fare Rule

y = 2x + 5

Distance ($x$)	Total Cost ($y$)
0 miles (Base Fee)	$5.00
5 miles	$15.00
10 miles	$25.00
Every extra mile adds exactly $2.00. That is the Slope!

Sytem of Linear Equations:

A system of linear equations is a collection of two or more linear equations that involve the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. When graphed, the solution to a system of two linear equations in two variables is the point (or points) where the lines intersect. If the lines are parallel, there's no solution; if they are the same line, there are infinitely many solutions. These systems are fundamental in mathematics and engineering for solving problems with multiple unknown quantities and constraints.

Understanding Systems of Linear Equations

A system of linear equations is simply a collection of two or more linear equations that share the same set of variables. The core challenge is to find a set of values for these variables that makes every single equation in the system true at the same time.

Example System:

2x + y = 7 x - y = 2

The Graphical Interpretation

When you plot each equation on a graph, the solution to the system is:

intersection Unique Solution

The lines cross at a single point. This point's coordinates (x, y) are the unique solution.

parallel_lines No Solution

The lines are parallel and never intersect. No point satisfies both equations.

line_start_circle Infinitely Many Solutions

The two equations actually represent the exact same line. Every point on that line is a solution.

Systems are crucial for solving problems with multiple unknown quantities and constraints in various fields like science, economics, and engineering.

Visualizing Systems of Equations

Unique Solution

The lines cross at exactly one point.

No Solution

The lines are parallel and will never touch.

Infinite Solutions

The lines sit on top of each other.

1. One Unique Solution{

                x + y = 5

                2x - y = 4
            
            Result: (3, 2)

            The lines intersect at a single point.
        
2. No Solution (Parallel){

                y = 2x + 3

                y = 2x - 1
            
            Result: None

            The slopes are the same, but the y-intercepts differ.
        
3. Infinite Solutions{

                x - 2y = 4

                2x - 4y = 8
            
            Result: All points on the line

            The second equation is just the first one multiplied by 2.

Matrix Equation:

In linear algebra, a system of equations can be represented as a Matrix Equation. This is a much more efficient way to handle large systems, especially when using computers or calculators.

The System of Equations

Let's take a simple 2D system:

$2x+3y=8$

$5x−y=3$

The Matrix Representation: AX=B

Equation to Matrix Conversion

2x + 3y = 8
5x - 1y = 3

2 3
5 -1

Matrix A (Coefficients)

x
y

Matrix X (Variables)

8
3

Matrix B (Constants)

Coefficient Matrix (A): Grabs the numbers in front of x and y.
Variable Matrix (X): Lists your unknowns in a column.
Constant Matrix (B): Contains the values from the right side of the equals sign.

Matrix Form: $AX = B$