Slope (or the steepness) of a line can be thought of as rise over run. Give, two points on the line,
\((x_1, y_1)\) and
\((x_2, y_2)\) then slope can be written as:
\(m=\frac{y_2-y_1}{x_2-x_1}\)
Note: No slope
\(\ne\) Zero slope
\(\ne\) Undefined slope. These all represent different things.
Every line has a slope, so the term
no slope does not make sense when referring to a line.
Zero slope refers to horizontal lines where
\(y_2=y_1\) so
\(m=0\text{.}\) Undefined slope refers to vertical lines where
\(x_2=x_1\) so
\(m\) is undefined.
Positive slope means that the line increases as
\(x\) increases or that the line rises to the right. Negative slope means that the line decreases as
\(x\) increases or that the line falls to the right
If \(m_1\) is the slope of line 1, and \(m_2\) is the slope of line 2. Then:
-
line 1 and line 2 are parallel if
\(m_1=m_2\text{.}\)
-
line 1 and line 2 are perpendicular if
\(m_1=-\frac{1}{m_2}\text{.}\)
Forms of Linear Equations
There are three main forms for the equation of a line.
Standard form is
\(Ax+BY=C\) where
\(A, B\text{,}\) and
\(C\) are integers and
\(A\gt 0\text{.}\)
The point-slope form is useful when you know the slope
\(m\) and a point,
\((x_1,y_1)\text{,}\) on the line. The point-slope form is given as
\(y-y_1=m(x-x1)\)
The slope-intercept form is most commonly used to graph a line. For this form, we need the slope
\(m\) and the y-intercept
\((0,b)\text{.}\) The slope-intercept form is given by
\(y=mx+b\text{.}\)
There are also two special cases for lines; horizontal and vertical lines, which correspond to our specific slopes. For a horizontal line
\(m=0\) so any form of the equation can be reduced to
\(y=b\) where
\((0,b)\) is the y-intercept. For a vertical line,
\(m\) is undefined. So any form of the equation, can be reduced to
\(x=a\) where
\((a,0)\) is the x-intercept.
