# 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 ${\displaystyle ax+by+c=0}$  where ${\displaystyle x}$ , ${\displaystyle y}$  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 ${\displaystyle y=ax+b}$  where ${\displaystyle x}$ , ${\displaystyle y}$  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

${\displaystyle y=3x+1}$

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

## Point-slope form

The Point-slope form take the form of ${\displaystyle y-y_{1}=m(x-x_{1})}$ , where ${\displaystyle x}$ , ${\displaystyle y}$  are variables, (${\displaystyle x_{1}}$ , ${\displaystyle y_{1}}$ ) 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

${\displaystyle y-3=4(x-2)}$

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, ${\displaystyle y=ax+b}$  and set ${\displaystyle x=0}$ , ${\displaystyle y=0}$  respectively. The result of this is two functions of variables ${\displaystyle x}$  and ${\displaystyle y}$  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:

${\displaystyle y=-5x+2}$

${\displaystyle y-3=4(x-2)}$

${\displaystyle y=4x+2}$

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

${\displaystyle 3y+4x-1=0}$