Slope of a Line in a Cartesian Plane

Slope of a Line in a Cartesian Plane

In a Cartesian plane, the slope m of the line that passes through two given points P(x1y1) and Q(x2y2) 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 = mx + 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 = 4x + 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 = 2x + 7 and y = 2x − 8 are parallel, because they have the same slope: 2.