The slope, $m$, of a line through the distinct points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is given by the formula $m=$ $\square$
Answer (2)
The slope of a line is defined as the change in the y-coordinate divided by the change in the x-coordinate.
Calculate the difference in y-coordinates: y 2 − y 1 .
Calculate the difference in x-coordinates: x 2 − x 1 .
Express the slope m as the ratio of these differences: m = x 2 − x 1 y 2 − y 1 .
Explanation
Understanding the slope The slope of a line passing through two distinct points ( x 1 , y 1 ) and ( x 2 , y 2 ) is defined as the change in the y-coordinate divided by the change in the x-coordinate. This represents the rate at which the line rises or falls as you move from left to right.
Calculating the slope To calculate the slope, we find the difference in the y-coordinates ( y 2 − y 1 ) and divide it by the difference in the x-coordinates ( x 2 − x 1 ). This gives us the formula: m = x 2 − x 1 y 2 − y 1 This formula holds true as long as x 1 = x 2 , because division by zero is undefined.
Final Answer Therefore, the slope, m , of a line through the distinct points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1
Examples
Imagine you're hiking up a mountain. The slope represents how steep the mountain is. If you know the elevation (y-coordinate) at two different points and the horizontal distance (x-coordinate) between those points, you can calculate the slope to determine the steepness of your climb. A larger slope means a steeper climb, while a smaller slope means a gentler climb. This concept is also used in construction to determine the steepness of roofs or ramps, ensuring they are safe and functional.
For example, if you are at 100 meters elevation at x=0 and 400 meters elevation at x=1000, then the slope is
m = 1000 − 0 400 − 100 = 1000 300 = 0.3
This means that for every 1 meter you move horizontally, you go up 0.3 meters in elevation.
The slope m of a line through points ( x 1 , y 1 ) and ( x 2 , y 2 ) is calculated using the formula m = x 2 − x 1 y 2 − y 1 . This represents the change in y divided by the change in x. It's essential to ensure that x 1 is not equal to x 2 to avoid division by zero.
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