Find the slope of the line that passes through the points (3,5) and (2,8).
Answer (1)
Recall the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the given points (3,5) and (2,8) into the formula.
Calculate the slope: m = 2 − 3 8 − 5 = − 1 3 = − 3 .
The slope of the line is − 3 .
Explanation
Understanding the Problem We are given two points, (3,5) and (2,8), and we want to find the slope of the line that passes through them. The slope of a line is a measure of its steepness and direction. It tells us how much the y-value changes for every unit change in the x-value.
Recalling the Slope Formula The formula to calculate the slope (m) of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by: m = x 2 − x 1 y 2 − y 1 This formula calculates the change in y divided by the change in x, which gives us the rate of change of the line.
Substituting the Values Let's assign the given points to the variables in the formula: x 1 = 3 , y 1 = 5 x 2 = 2 , y 2 = 8 Now, substitute these values into the slope formula: m = 2 − 3 8 − 5
Calculating the Slope Now, let's calculate the slope: m = − 1 3 = − 3 So, the slope of the line passing through the points (3,5) and (2,8) is -3.
Final Answer The slope of the line passing through the points (3,5) and (2,8) is − 3 . This means that for every 1 unit increase in the x-value, the y-value decreases by 3 units. The negative sign indicates that the line is decreasing as we move from left to right.
Examples
Imagine you're hiking on a trail, and you want to describe how steep the path is. If you know the elevation at two different points along the trail, you can use the slope formula to calculate the steepness. For example, if you climb from an elevation of 500 feet to 800 feet over a horizontal distance of 1000 feet, the slope of the trail is (800-500)/(1000-0) = 300/1000 = 0.3. This tells you that for every 10 feet you walk horizontally, you gain 3 feet in elevation. Understanding slope helps you quantify and communicate the steepness of paths, roads, or any linear incline.
