Understanding and Calculating the Slope of a Line Between Given Points

When dealing with the geometry of a line that passes through two points, the concept of slope plays a pivotal role. Slope, denoted as ( m ), represents the change in the vertical distance (rise) over the change in the horizontal distance (run) between two points. This article will walk you through the process of calculating the slope of the line joining points (left(frac{8}{3}, frac{1}{3}right)) and (left(frac{5}{4}, frac{13}{4}right)).

Step-by-Step Guide to Calculating the Slope

The formula for the slope ( m ) of a line joining two points ((x_1, y_1)) and ((x_2, y_2)) is given by:[ m frac{y_2 - y_1}{x_2 - x_1} ]

Let's plug in our given points into this formula and solve it step by step.

Step 1: Calculate the Numerator

The numerator is ( y_2 - y_1 ). Given the coordinates (left(frac{8}{3}, frac{1}{3}right)) and (left(frac{5}{4}, frac{13}{4}right)), we have:[ y_2 - y_1 frac{13}{4} - frac{1}{3} ]To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 3 is 12. Converting both fractions to have a denominator of 12:[ frac{13}{4} frac{13 times 3}{4 times 3} frac{39}{12} ][ frac{1}{3} frac{1 times 4}{3 times 4} frac{4}{12} ]Subtracting these fractions:[ frac{39}{12} - frac{4}{12} frac{39 - 4}{12} frac{35}{12} ]

Step 2: Calculate the Denominator

The denominator is ( x_2 - x_1 ). Given the coordinates, we have:[ x_2 - x_1 frac{5}{4} - frac{8}{3} ]Again, we need a common denominator. The LCM of 4 and 3 is 12. Converting both fractions:[ frac{5}{4} frac{5 times 3}{4 times 3} frac{15}{12} ][ frac{8}{3} frac{8 times 4}{3 times 4} frac{32}{12} ]Subtracting these fractions:[ frac{15}{12} - frac{32}{12} frac{15 - 32}{12} frac{-17}{12} ]

Step 3: Calculate the Slope

Now, we substitute the values we found into the slope formula:[ m frac{frac{35}{12}}{frac{-17}{12}} frac{35}{12} times frac{12}{-17} frac{35}{-17} -frac{35}{17} ]Therefore, the slope of the line joining the points (left(frac{8}{3}, frac{1}{3}right)) and (left(frac{5}{4}, frac{13}{4}right)) is (boxed{-frac{35}{17}}).

Summary and Check

Through careful calculation, we found that the slope of the line joining (left(frac{8}{3}, frac{1}{3}right)) and (left(frac{5}{4}, frac{13}{4}right)) is (-frac{35}{17}). This result can be cross-verified by recalculating the steps, ensuring the solution is accurate.

This method can be applied to similar problems involving the calculation of slope between any two points, making use of the common denominator technique to simplify fraction operations.