Linear Equations and Systems

1. Introduction to Linear Equations

A linear equation is an equation that represents a straight line (in two dimensions), a plane (in three dimensions), or a hyperplane (in higher dimensions). We classify them by how many variables they have.

Linear Equations in 1 Variable ($x$)

These are written in the standard form:

(where $a$ and $b$ are constants)

When trying to solve for $x$, there are exactly three possible outcomes:

Exactly One Solution: If $a \neq 0$, you can divide by $a$ to get $x = \frac{b}{a}$.

No Solution: If $a = 0$ and $b \neq 0$ (e.g., $0x = 3$). No number times zero will ever equal a non-zero number!

Infinitely Many Solutions: If $a = 0$ and $b = 0$ (e.g., $0x = 0$). Any real number will work here. The solution set is all real numbers, denoted as $S = \mathbb{R}$.

Linear Equations in 2 Variables ($x, y$)

These take the form:

(where $a, b \neq 0$ simultaneously)

Because we have one equation and two variables, we cannot find a single numerical answer. Instead, we have infinitely many solutions. To solve this, we use a parameter.

General Solution: We fix one variable to be a parameter (like $t$). For example, in $2x + y = 7$, we can let $x = t$. Solving for $y$ gives $y = 7 - 2t$. Our general solution set is $(x, y) = (t, 7 - 2t)$ for any real number $t$.

Particular Solution: If we plug in a specific number for $t$, we get a specific point. If $t = 1$, the particular solution is $(1, 5)$.

Linear Equations in 3 or More Variables

We follow the exact same logic for 3 variables ($ax + by + cz = d$) or $n$ variables ($a_1x_1 + a_2x_2 + \dots + a_nx_n = b$). To solve a single equation with $n$ variables, you will need to assign parameters to $n - 1$ of those variables and solve for the last remaining one.

2. Systems of Linear Equations

A system of linear equations is simply a finite set of linear equations that we want to solve at the same time.

Fundamental Theorem

Every linear system has exactly one of the following:

• No solutions
• Exactly one solution
• Infinitely many solutions

The Augmented Matrix

Writing out $x, y$, and $z$ over and over can get messy. To make solving systems cleaner and more efficient, mathematicians use an augmented matrix. We drop the variables and the equals signs, and just write the constants (coefficients) in a grid, separated by a vertical line.

For example, this system:

Becomes this augmented matrix:

$$\left[\begin{array}{ccc|c} 1 & 1 & 4 & 2 \\ 1 & -3 & 1 & 5 \end{array}\right]$$