 The equation of a line

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Assume that $$a$$, $$b$$, and $$c$$ are constant real numbers: parameters.

The solution to the equation $$a\cdot x+b\cdot y+c=0$$ can be drawn in the plane. They are the points $$\left[x,y\right]$$ satisfying $$a\cdot x+b\cdot y+c=0$$.  If $$a\ne0$$ or $$b\ne0$$, then these points form a straight line, or simply just a line.

• If $$b\ne0$$, then the equation can be written as $$y=-\frac{a}{b}\cdot{x}-\frac{c}{b}$$. For, these are the solutions if we consider $$x$$ as a parameter and $$y$$ as unknown. This indicates that for every value of $$x$$ there is a point $$\left[x,y\right]$$ with $$y$$ equal to $$-\frac{a}{b}\cdot{x}-\frac{c}{b}-\frac{a}{b}\cdot{x}-\frac{c}{b}$$.
• If $$a\ne0$$, the line is oblique (by oblique we mean neither horizontal nor vertical).
• If $$a = 0$$, then the value of $$y$$ is constant, equal to $$-\frac{c}{b}$$. In this case the line is horizontal.
• In the exceptional case $$b = 0$$ the equation looks like $$a \cdot{x}+c = 0$$.
• If $$a\ne0$$, then the line is vertical.
• If $$a = 0$$ and
• $$c\ne0$$, then there are no solutions
• $$c = 0$$, then each pair of values of $$\left[x,y\right]$$ is a solution.

A straight line can be described in different ways.

1. The solutions $$\left[x,y\right]$$ to an equation $$a\cdot x+b\cdot y+c=0$$ with unknowns $$x$$ and $$y$$. Here $$a$$, $$b$$ and $$c$$ are real numbers such that $$a$$ and $$b$$ are not equal to zero.
2. The line through two given points in the plane: if $$P = \left[x,y\right]$$ and $$Q = \left[s,t\right]$$ are points in the plane, then the line through $$P$$ and $$Q$$ has equation $$a\cdot x+b\cdot y+c=0$$ with $$a=q-t$$, $$b = s – p$$ and $$c = t \cdot {p} – q \cdot{s}$$.
3. The line through a given point, the base point, and a direction, indicated by the number $$-\frac{a}{b}$$, where $$a$$ and $$b$$ are as in the equation given above; this number is called the slope of the line.
4. The line with function representation $$y = p\cdot x+q$$ if $$b\ne0$$ and $$x = r$$ otherwise; here we have $$p = -\frac{a}{b}$$ (the slope), $$q = -\frac{c}{b}$$ (the intercept), which is the value of $$y$$ for $$x = 0$$ and $$r = -\frac{c}{a}$$ in terms of the above $$a$$, $$b$$ and $$c$$. This can be seen as a special case of the previous description, with base point $$\left[0,q\right]$$. In the case where $$b\ne0$$, the variable $$y$$ is a function of $$x$$, in the other case, $$x$$ is a constant function of $$y$$.

Let’s practice!

The line segment between the two points $$\left[2,\frac{77}{12}\right]$$ and $$\left[6,\frac{63}{4}\right]$$ is drawn in the figure below. Give the function rule for this line in the form $$y = a\cdot{x} + b$$.

The line is described by the function rule $$y = a\cdot{x} + b$$ where $$a$$ is the slope, given as the quotient of the difference of the $$y$$-values with the difference of the $$x$$-values of two points on the line. Hence $$a=\frac{{{63}\over{4}}-{{77}\over{12}}}{6-2}={{7}\over{3}}$$. The value of $$b$$ follows from $$b = y – a\cdot{x}$$, where $$\left[x,y\right]$$ is a random point on the line. Hence $$b={{77}\over{12}}-{{7}\over{3}}\cdot{2}={{7}\over{4}}$$.

$$y={{7}\over{3}}\cdot{x}+{{7}\over{4}}$$