An equation of a line in the plane

Solving linear equations and inequalities: An equation of a straight line in a plane

An equation of a line in the plane

Equation of a line in the plane

Let $x, y$ be variables and $a, b, c$ be parameters such that $a\neq 0$ or $b\neq 0$ .

The solutions of the linear equation $ax+by+c=0$ form a straight line $\ell$ in the $x$ - $y$ plane.

If $a=0$ , then the line $\ell$ is horizontal.
If $b=0$ , then the line $\ell$ is vertical.
If $c=0$ , then the line $\ell$ goes through the origin.

The ratio $a:b$ determines the direction of the line.
The ratio $c:a$ determines the intercept with the $x$ axis.
The ratio $c:b$ determines the intercept with the $y$ axis.

GeoGebra

Actually we speak about an equation of a line because more equations are possible: for each number $k\ne 0$ the equations $ax+by+c=0$ and $(ka)x+(kb)y+(kc)=0$ describe the same line. In other words, the 3-tuples $\rv{a,b,c}$ and $\rv{ka,kb,kc}=k\rv{a,b,c}$ describe the same line. Two 3-tuple that are equal to each other up to a scalar multiplication are called equivalent and an equivalence class of vectors under this equivalence relation is called a homogeneous vector. Every row vector $\rv{a,b,c}$ is a representative of the equivalence class and describes the line gven by the equation $ax+by+c=0$ . You can also describe points in a plane with homogeneous coordinates: $\rv{x,y,1}$ is the homogeneous vector that describes the point $(x,y)\in\mathbb{R}^2$ ; other homogeneous coordinates of this point are for example $\rv{2x,2y,2}$ and $\rv{-x,-y,-1}$ .

When a line in the plane is not vertical, then it can also be described by a linear formula. This linear formula can be deduced from the above general equation by isolating the variable $y$ .

Let $x, y$ be variables and $r,s$ be parameters. Then, the linear formula is $y=rx+s$ a description of a line $\ell$ in the $x$ - $y$ plane.

The parameter $r$ is called the slope or gradient
and determines the direction of the line.

If $r>0$ we have a graph that slopes upwards from left to right; $r=0$ means a horizontal line; if $r<0$ we have a graph that slopes downwards from left to right.

With a horizontal increase $\Delta x$ we get a vertical increase $\Delta y$ of $r\cdot \Delta x$ . So $r=\frac{\Delta y}{\Delta x}$ .

If $\Delta x=1$ , then the vertical shift equals $r$ . In other words, if we move at a point on the line $1$ to the right, then we go $r$ up and end again on the line.

The parameter $s$ is the $y$ intercept: the line $\ell$ intersects the $y$ axis in the point $(0,s)$ .

GeoGebra