The equation of a line can be formed by looking at its graph. If two points on a line are known, the equation can be determined without a graph.

Example 1

Determine the equation of the line in the picture.

The equation of the line is of the form [[$ y = kx + b $]]. The constant term [[$ b $]] is obtained from the intersection of the line and the [[$ y $]]-axis, in this case [[$ b = 2 $]]. The slope factor [[$ k $]] is calculated as follows:

[[$ \displaystyle\frac {y \text {-axial change}} {x \text {-axial change}} = \: \displaystyle\frac {–2} {6} = \: –\displaystyle\frac {1} {3} $]]​

Answer: The equation of the line is [[$ y = – \displaystyle\frac {1} {3}x + 2 $]]​.

Example 2

A line passes through the points [[$ (-1, -6) $]] and [[$ (2, 0) $]]. Determine the equation of the line without drawing a graph.

Use the given points to calculate the slope of the line:

Thus, the equation of the line is now of the form [[$ y = 2x + b $]]. To solve the constant term [[$ b $]], place the values of one of the points inside the expression and solve the resulting equation for the variable [[$ b $]]. Select [[$ (-1, -6) $]] as the viewpoint.

[[$ \begin{equation} \label{eq1} \begin{split} -6 & = 2 · (-1) + b \\ -6 & = -2 + b \\ -b & = -2 + 6 \\ -b & = 4 \; \;  \; \; \; \; \;  \; \; \; \; \; \;  \; \; \; ||:(–1) \\ b & = -4 \\ \end{split} \end{equation} $]]​

Answer: The equation of the line is [[$ y = 2x - 4 $]]​.

NB! The equation of the line in its general form is [[$ 2x - y - 4 = 0 $]]​.