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

Homogeneous description of a point and line in a plane 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 \(\rv{-x,-y,-1}\).

Alternative equation for a nonvertical line 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