In a Cartesian plane, the slope

*m*of the line that passes through two given points P(*x*_{1},*y*_{1}) and Q(*x*_{2},*y*_{2}) is the ratio of the variation of the y-coordinates to the variation of the x-coordinates.The concept of slope is related to the study of figures in the Cartesian plane, in which the vector space is orthonormal.

The term *slope* is sometimes replaced by the expressions *highest degree coefficient* or *slope angle*, in some texts.

### Formula

The formula to calculate the slope *m* of a line that passes through the points P P(\(x_{1}\), \(y_{1}\)) and Q(\(x_{2}\), \(y_{2}\)) is: \(m = \dfrac{∆y}{∆x}\) = \(\dfrac{y_2\space –\space y_1 }{x_2\space –\space x_1 }\), where ∆y represents the variation of the y-coordinates and ∆x represents the variation of the x-coordinates.

### Properties

- If the line is not parallel to the y-axis, its equation is in the form
*y*= m*x*+ b where*m*represents the slope of the line. - In road signs, slope is expressed as a percentage. Therefore, a slope of 10% means that over a horizontal distance of 100 metres, the difference in height is 10 m, like for example:

\(\frac{410\space –\space 400}{100}\) = \(\frac{10}{100}\) = 10 %. - In a Cartesian plane, two lines with the same slope are parallel, and vice versa.
- In a Cartesian plane, two perpendicular lines have inverse slopes and opposite signs and the product of their slopes is equal to –1.

### Examples

- The line of the equation
*y*= 4*x*+ 3 has a slope of 4. - The slope of the line that passes through the points P(−3, 3) and Q(6, −1) is given by:

*m*= \(\dfrac{∆y}{∆x}\) = \(\dfrac{−1 –3}{6 –(−3)} = \dfrac{−4}{9}\). - The lines of the equations
*y*= 2*x*+ 7 and*y*= 2*x*− 8 are parallel, because they have the same slope: 2.