Linear Equations

# Linear Equations

 Definition: An equation in the form \$a_0 + a_1x_1 + a_2x_2 + ... + a_nx_n = 0\$ is called a Linear Equation of the \$n\$-variables \$x_1, x_2, ..., x_n\$.

If we are dealing with equations of two variables, we will often use \$x\$ and \$y\$ instead of \$x_1\$ and \$x_2\$. For example, the equation \$2 + 3x + 4y = 0\$ is a linear equation. If we are dealing with equations of three variables, we will often use \$x\$, \$y\$ and \$z\$ instead of \$x_1\$, \$x_2\$, and \$x_3\$. For example, the equation \$1 + 3x - 5y + 4z = 0\$ is a linear equation. If we are dealing with equations of \$n\$-many variables, we will often use the former notation. For example, the equation \$2 + 3x_1 + x_2 - 4x^3 + 0x^5 = 0\$ represents a linear equation of \$5\$ variables (although the variable \$x^4\$ is not present in this linear equation).

We will now review what linear equations of 2 variables and 3 variables represent geometrically.

## Linear Equations of 2-Variables

A linear equation of 2 variables can be conventionally written in the form \$a_1x_1 + a_2x_2 = b\$, however, the form \$y = mx + b\$ where \$m\$ is the slope of the line formed from the equation and \$b\$ is the \$y\$-intercept is commonly used as well. 2-variable linear equations form lines on a 2-dimensional coordinate system (typically the \$xy\$-coordinate or Cartesian coordinate system). For example, the following graph depicts the equation \$x - y = -2\$:

## Linear Equations of 3-Variables

A linear equation of 3 variables written in the form \$a_1x_1 + a_2x_2 + a_3x_3 = b\$ or \$ax + by + cz = d\$ represents a plane in 3-dimensional space, typically an \$xyz\$-coordinate system. For example, the following graph depicts the linear equation \$x + y + 3z = 2\$:
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://mathonline.wikidot.com/linear-equations", "fetch_time": 1721534594000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-30/segments/1720763517550.76/warc/CC-MAIN-20240721030106-20240721060106-00456.warc.gz", "warc_record_offset": 21401222, "warc_record_length": 5074, "token_count": 532, "char_count": 1781, "score": 4.84375, "int_score": 5, "crawl": "CC-MAIN-2024-30", "snapshot_type": "latest", "language": "en", "language_score": 0.8525665998458862, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00028-of-00064.parquet"}