Linear Equations

In essence, Linear Equations are equations that represent straight lines. They are the simplest form of equations and are used to represent relationships between variables. The general form of a linear equation is outlined as follows.

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What is a Linear Equation

$$a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n=b$$

• the $a_n$ variables serve as coefficients
• the $x_n$ variables serve as variable irrationals
• $b$ is a constant

However, there isn’t that much interesting about a single linear equation. Instead, what we will be focusing on are Systems of Linear Equations.

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Systems of Linear Equations

$$\begin{array}{r}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=b_1\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=b_2\\ ⋮⋮⋮\\ a_{51}{x}_1+a_{52}{x}_2+\cdots +a_{5n}{x}_n=b_5\end{array}$$

Definition

A system of linear equations is a list of linear equations that all have the same variables

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Example

Solving a System of Linear Equations

$$\begin{array}{rl}3x-2y+z& =5\\ -x+0y+2z& =-1\\ 0x+3y-z& =-3\end{array}$$

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Definition

A solution to a system of linear equations is a list of numbers that make all the equations true when substituted for the variables.

Question

What is a solution to the equation presented in the Example?

Let’s try a ballpark guess

$$\begin{array}{rl}y& =-1\\ x& =1\\ z& =0\end{array}$$

While we think this is an answer, all we have really done is just make another system of linear equations.

While some linear equations like the example can possibly be solved by “eyeballing it”, we need a more refined way of finding an answer to these problems.

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Elementary Operations

There are 3 elementary operations we should be aware of:

1. Swap
   • We exchange one equation with another
   • $R_i\leftarrow \to R_j$
2. Scale
   • We multiply the constant and the coefficients by a nonzero number.
   • $s{R}_j$
3. Combo
   • We add a multiple of one equation to another
   • $t{R}_i+R_j$

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Another Example

Example

$$\begin{array}{rl}2x+2y-2z& =2\\ x-2y+z& =-1\\ 2x-3y-3z& =1\end{array}$$

With the help of the elementary operations, we get:

applying $-2R_2+R_1,-2R_2+R_3$

$$\begin{array}{rl}7y-4z& =4\\ x-2y+z& =-1\\ y-5z& =3\end{array}$$

applying $-7R_3+R_1,2R_3+R_2$

$$\begin{array}{rl}31z& =-17\\ x-9z& =5\\ y-5z& =3\end{array}$$

Solution

We get an answer of:

$$\frac{2}{31},\frac{8}{31},-\frac{17}{31}$$