How to find the slope with fractions?
Understand the Problem
The question is asking for a method or procedure to calculate the slope of a line when the numbers involved are fractions. The slope is typically calculated as the difference in the y-values divided by the difference in the x-values (rise over run), and this can be applied even when using fractional values.
Answer
The slope of the line is $m = \frac{1}{3}$.
Answer for screen readers
The slope of the line is $m = \frac{1}{3}$.
Steps to Solve
-
Identify Points on the Line Determine the two points on the line, which can be represented as $(x_1, y_1)$ and $(x_2, y_2)$. For example, let's say the points are $(\frac{1}{2}, \frac{3}{4})$ and $(\frac{3}{4}, \frac{5}{6})$.
-
Calculate the Differences Compute the differences between the y-values and x-values of the two points. For y-values: $$ \Delta y = y_2 - y_1 $$ For x-values: $$ \Delta x = x_2 - x_1 $$
-
Substitute the Values Substitute the fractional values into the equations from the previous step. For our example: $$ \Delta y = \frac{5}{6} - \frac{3}{4} $$ $$ \Delta x = \frac{3}{4} - \frac{1}{2} $$
-
Find a Common Denominator To subtract these fractions, find a common denominator. For our example, the least common multiple of 6 and 4 is 12. Thus: $$ \Delta y = \frac{5}{6} \to \frac{10}{12} $$ $$ \Delta y = \frac{3}{4} \to \frac{9}{12} $$ Now perform the subtraction: $$ \Delta y = \frac{10}{12} - \frac{9}{12} = \frac{1}{12} $$
-
Calculate the Change in x-values Now repeat this for the x-values: $$ \Delta x = \frac{3}{4} \to \frac{9}{12} $$ $$ \Delta x = \frac{1}{2} \to \frac{6}{12} $$ Now perform the subtraction: $$ \Delta x = \frac{9}{12} - \frac{6}{12} = \frac{3}{12} = \frac{1}{4} $$
-
Calculate the Slope Finally, calculate the slope $m$ using the formula: $$ m = \frac{\Delta y}{\Delta x} $$ Substituting in our values: $$ m = \frac{\frac{1}{12}}{\frac{1}{4}} $$ To divide fractions, multiply by the reciprocal: $$ m = \frac{1}{12} \times \frac{4}{1} = \frac{4}{12} = \frac{1}{3} $$
The slope of the line is $m = \frac{1}{3}$.
More Information
The slope of a line is a measure of its steepness and direction. A positive slope indicates that as you move to the right on the x-axis, the y-values increase, while a negative slope shows the opposite. The slope can be calculated using fractional points, just as with whole numbers, showing that the concepts of rise and run apply universally.
Tips
- Forgetting to find a common denominator when subtracting fractions. Always ensure you have the same denominator before subtracting.
- Misapplying the formula for slope; remember to order points correctly as $(x_1, y_1)$ and $(x_2, y_2)$.
