A variety of branches of mathematics use slope. In geometry, you can use the slope to plot points on a line, including lines that define the shape of a polygon. Statisticians use slope to describe the correlation between two variables. Economists use slope to show and predict rates of change. People also use slope in real, concrete ways. For example, slope is used when constructing roads, stairs, ramps, and roofs.
For example a slope of a line might be 21{\displaystyle {\frac {2}{1}}}. This means that to go from one point to the next, you need to go up 2 along the y-axis, and over 1 along the x-axis.
For example, in the equation y=3x+1{\displaystyle y=3x+1}, the slope would be 3{\displaystyle 3}. You can still think of this slope in terms of rise over run if you turn it into a fraction. Any whole number can be turned into a fraction by placing it over 1. So, 3=31{\displaystyle 3={\frac {3}{1}}}. This means that the line represented by this equation rises 3 units vertically for every 1 unit it runs horizontally.
For example, a slope of 2 (that is, 21{\displaystyle {\frac {2}{1}}}) is steeper than a slope of 0. 5 (12{\displaystyle {\frac {1}{2}}}).
A positive slope is denoted by a positive number.
A negative slope is denoted by a negative number, or a fraction with a negative numerator. To help remember the difference between a positive and negative slope, you can think of yourself as standing on the left endpoint of the line. If you need to walk up the line, it’s positive. If you need to walk down the line, it’s negative. [7] X Research source Knowing the difference between negative and positive slopes can help you check that your calculations are reasonable.
For example, you might choose the points (4, 4) and (12, 8).
Your rise will be negative if you start with the higher point and move down to the lower point. For example, beginning at the point (4, 4), you would count up 4 positions to point (12, 8). So, the rise of your slope is 4: slope=4run{\displaystyle {\text{slope}};={\frac {4}{\text{run}}}}.
Your run will be negative if you start with the point on the right and move over to the left. For example, beginning at the point (4, 4), you would count over 8 positions to point (12, 8). So, the run of your slope is 8: slope=48{\displaystyle {\text{slope}};={\frac {4}{8}}}.
For example, 4 and 8 are both divisible by 4, so the slope 48{\displaystyle {\frac {4}{8}}} simplifies to 12{\displaystyle {\frac {1}{2}}}. Note that it is a positive slope, so the line moves up to the right.
For example, if your points are (-4, 7) and (-1, 3), your formula will look like this: m=3−7−1−(−4){\displaystyle m={\frac {3-7}{-1-(-4)}}}.
For example:m=3−7−1−(−4){\displaystyle m={\frac {3-7}{-1-(-4)}}}m=−43{\displaystyle m={\frac {-4}{3}}}So, the slope of the line is −43{\displaystyle {\frac {-4}{3}}}. Note that since the slope is negative, the line is moving down to the right.
