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.

equation of a line

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 \(b={{77}\over{12}}-{{7}\over{3}}\cdot{2}={{7}\over{4}}\).




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