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 formEdit

The General form is shown as   where  ,   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 formEdit

The Slope-intercept form take the form of   where  ,   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.

ExampleEdit

 

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

Point-slope formEdit

The Point-slope form take the form of  , where  ,   are variables, ( ,  ) 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.

ExampleEdit

 

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

Solving linear equations / GraphingEdit

The most direct way of graphing a linear equation is to use its Slope-intercept form,   and set  ,   respectively. The result of this is two functions of variables   and   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 SetEdit

Graph the following linear equations:Edit