Slope

Slope Between Two Points

Given two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$, the slope between these two points is $m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$. That is, the slope between two points is the difference between the y-coordinates of the points, divided by the difference between the x-coordinates of the points. For example, the slope between the two points (1, 3) and (5, 6) is ${\frac {6-3}{5-1}}={\frac {3}{4}}$. The slope between (-1, -1) and (15, -21) is ${\frac {-21-(-1)}{15-(-1)}}={\frac {-21+1}{15+1}}=-{\frac {20}{16}}=-{\frac {5}{4}}$.

Point-Slope Form of a Line

The equation for a line with a slope of $m$ that goes through some point $(x_{1},y_{1})$, in point-slope form, is $y-y_{1}=m(x-x_{1})$. For example, the equation of a line with a slope of 3 that goes through the point (1, 4) is $y-4=3(x-1)$. The equation of a line with a slope of $-{\frac {1}{2}}$ that goes through point (-7, -7) is $y+7=-{\frac {1}{2}}(x+7)$.

Slope-Intercept Form of a Line

Another form of an equation of a line is slope-intercept form. The equation of a line with a y-intercept of b (that is, a line that intersects the y-axis at the point (0, b)) and a slope of m is $y=mx+b$. For example, the equation of a line with a y-intercept of 5 and slope of 6 is $y=5x+6$. Note that this equation is equivalent to point-slope form. A line with a y-intercept of 5 goes through the point (0, 5), so the point-slope form of this same line is $y-5=6(x-0)=6x$. By adding 5 to each side of the equation, we get the slope-intercept form of the line.

Resources

• Slope Between Points, Illustrative Mathematics
• Forms of Linear Equations, Khan Academy