\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Line Equations

Definition


An equation like \(\textcolor{colorprop}{y}=2\textcolor{colordef}{x}-1\) describes a relationship between the variables \(\textcolor{colordef}{x}\) and \(\textcolor{colorprop}{y}\). For any value of \(\textcolor{colordef}{x}\) we choose, the equation tells us the corresponding value of \(\textcolor{colorprop}{y}\).
We can use this to find coordinates \((\textcolor{colordef}{x}, \textcolor{colorprop}{y})\) for points that satisfy the equation.
  • If \(\textcolor{colordef}{x} = \textcolor{colordef}{1}\), then \(\textcolor{colorprop}{y} = 2(\textcolor{colordef}{1}) - 1 = \textcolor{colorprop}{1}\). This gives us the point \((1, 1)\).
  • If \(\textcolor{colordef}{x} = \textcolor{colordef}{2}\), then \(\textcolor{colorprop}{y} = 2(\textcolor{colordef}{2}) - 1 = \textcolor{colorprop}{3}\). This gives us the point \((2, 3)\).
Let's build a table of values to find more points:
\(\textcolor{colordef}{x}\) \(\textcolor{colordef}{0}\) \(\textcolor{colordef}{1}\) \(\textcolor{colordef}{2}\) \(\textcolor{colordef}{3}\)
\(\textcolor{colorprop}{y}\) \(\textcolor{colorprop}{-1}\) \(\textcolor{colorprop}{1}\) \(\textcolor{colorprop}{3}\) \(\textcolor{colorprop}{5}\)
Each pair in the table, for example \((0,-1)\) or \((3,5)\), makes the equation \(\textcolor{colorprop}{y}=2\textcolor{colordef}{x}-1\) true. When we plot these points on a coordinate plane, we see they all lie on the same straight line. The equation \(\textcolor{colorprop}{y}=2\textcolor{colordef}{x}-1\) is the equation of this line, because it is true for every point on the line (and only for the points on this line).

Definition Slope-Intercept Form
The slope-intercept form of a line's equation is:$$y = mx + c$$where \(m\) is the slope (gradient) and \(c\) is the \(y\)-intercept.
  • The slope \(m\) tells us how much \(y\) changes when \(x\) increases by \(1\).
  • The \(y\)-intercept \(c\) is the value of \(y\) when \(x = 0\), i.e. the point where the line crosses the \(y\)-axis.
Example
This line has slope \(m=-2\) and \(y\)-intercept \(c=1\).