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

The equation for a line with a slope of $m$ that goes through some point $(x_{1},y_{1})$, in point-slope form, is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - 4 = 3(x - 1) } . The equation of a line with a slope of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{1}{2} } that goes through point (-7, -7) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y + 7 = -\frac{1}{2}(x + 7) } .

Slope-Intercept Form

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = mx + b } . For example, the equation of a line with a y-intercept of 5 and slope of 6 is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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