Finding the slope of a line is a fundamental concept in algebra that helps us understand how a line behaves on a graph. The slope indicates the steepness of the line and the direction it travels. In algebra, the slope is calculated as the change in the y-values divided by the change in the x-values between two distinct points on the line. Let’s explore three practical examples of finding the slope of a line in different contexts.
Imagine you’re tracking a hiker climbing a mountain. The hiker starts at the base of the mountain, which we can represent as the point (2, 3), where 2 is the horizontal distance (x) and 3 is the vertical height (y). After some time, the hiker reaches a point (6, 11) higher up the mountain.
To find the slope, we use the formula:
Slope (m) = (y2 - y1) / (x2 - x1)
Here, (x1, y1) = (2, 3) and (x2, y2) = (6, 11).
Substituting in the values:
m = (11 - 3) / (6 - 2)
m = 8 / 4 = 2
The slope of the line representing the hiker’s climb is 2. This means for every 1 unit the hiker moves horizontally, they gain 2 units in height.
Note: This slope can indicate a steep climb. If the slope were less than 1, it would suggest a gentler incline.
Consider a car traveling from a starting point to a destination. Let’s say the car leaves a town at coordinates (1, 1) and reaches a destination in a neighboring town at (4, 7).
To find the slope of the line representing the car’s journey, we again use the slope formula:
Slope (m) = (y2 - y1) / (x2 - x1)
Using our points: (x1, y1) = (1, 1) and (x2, y2) = (4, 7):
m = (7 - 1) / (4 - 1)
m = 6 / 3 = 2
The slope of 2 indicates that for every 1 unit the car moves forward, it ascends 2 units vertically. This could represent an uphill road.
Variation: If the car had traveled from (4, 7) back to (1, 1), the slope would still be 2, but it would be interpreted as descending if we think of the journey in reverse.
Let’s say you’re analyzing the price of a stock over a week. On Monday, the stock price was \(10 (point A: (0, 10)), and by Friday, it has risen to \)20 (point B: (5, 20)).
Using the slope formula:
Slope (m) = (y2 - y1) / (x2 - x1)
We have (x1, y1) = (0, 10) and (x2, y2) = (5, 20):
m = (20 - 10) / (5 - 0)
m = 10 / 5 = 2
This slope of 2 can indicate that the stock price increased by $2 for every day of the week, reflecting a consistent upward trend in the stock market.
Relevance: Understanding the slope in this financial context is crucial for making investment decisions, as it helps predict future price movements.