Linear algebra/Linear equations

Linear equations, also known as first-degree equations, are algebraic equations that contain a variable which always has the highest power of 1. There are multiple ways a linear equation can be presented in including the General form, the Slope-intercept form and Point-slope form. The different forms of presentation emphasizes different aspects of each equation. The graph of a linear equation on the xy-plane is always a line, thus the name "linear".

General form

The General form is shown as $ax+by+c=0$ where $x$, $y$ are variables and a,b as constants (not both 0). This provides the possibility of easily converting to other forms of the equation for different purposes.

Slope-intercept form

The Slope-intercept form take the form of $y=ax+b$ where $x$, $y$ are variables and a,b as constants. This form emphasizes the slope of the linear equation along with the y-intercept of the equation represented by constants a and b respectively. Intuitively, the y-intercept of an equation is the y-value for when the graph intersects the y-axis.

Example

$y=3x+1$

The slope of the equation is 3 with the y-intercept being 1.

Point-slope form

The Point-slope form take the form of $y-y_{1}=m(x-x_{1})$, where $x$, $y$ are variables, ($x_{1}$, $y_{1}$) is a point on the equation, and m is the slope of the equation. This form points out a point on the equation while also showing the slope.

Example

$y-3=4(x-2)$

The slope of the equation is 4 with (2,3) being a point that the equation passes through.

Solving linear equations / Graphing

The most direct way of graphing a linear equation is to use its Slope-intercept form, $y=ax+b$ and set $x=0$, $y=0$ respectively. The result of this is two functions of variables $x$ and $y$ set equal to constants. Solving for the variables will result in two points on the xy-axis where the equation intercepts, connecting the two points with a straight line will result in the graph of the linear equation.

Problem Set

Graph the following linear equations:

$y=-5x+2$

$y-3=4(x-2)$

$y=4x+2$

$y+1={1 \over 2}(x+1)$

$3y+4x-1=0$