Find the slope of the line that passes through the points $(3,5)$ and $(2,8)$.
Answer (1)
Identify the coordinates of the two given points.
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the given coordinates into the formula and simplify.
The slope of the line is: − 3 .
Explanation
Problem Analysis We are given two points, ( 3 , 5 ) and ( 2 , 8 ) , and we want to find the slope of the line that passes through them.
Slope Formula The slope of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 In our case, we have ( x 1 , y 1 ) = ( 3 , 5 ) and ( x 2 , y 2 ) = ( 2 , 8 ) .
Calculate the Slope Substitute the coordinates of the given points into the slope formula: m = 2 − 3 8 − 5 = − 1 3 = − 3 So, the slope of the line is − 3 .
Final Answer The slope of the line passing through the points ( 3 , 5 ) and ( 2 , 8 ) is − 3 .
Examples
Understanding the slope is crucial in many real-world applications. For instance, if you're analyzing the relationship between the number of hours studied and the score on a test, the slope tells you how much the score increases for each additional hour of studying. Similarly, in construction, the slope of a ramp determines its steepness, affecting its usability. Knowing how to calculate slope helps in making informed decisions in various fields.
