Find The Slope Of Line AB Given Points A And B

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Let's dive into finding the slope of a line when we're given two points on that line. This is a fundamental concept in coordinate geometry, and understanding it opens doors to solving various problems related to lines and their properties. In this case, we're given two points, A(4, 5) and B(9, 7), that lie on line AB. Our mission is to determine the slope of this line. So let's break down the process step by step.

Understanding Slope

Before we calculate anything, let's quickly recap what slope actually means. Slope, often denoted by the letter 'm', describes the steepness and direction of a line. It tells us how much the y-value changes for every unit change in the x-value. A positive slope indicates that the line is increasing (going upwards) as you move from left to right, while a negative slope indicates a decreasing line (going downwards). A slope of zero means the line is horizontal, and an undefined slope means the line is vertical.

Slope is often described as "rise over run," where:

  • Rise is the vertical change between two points.
  • Run is the horizontal change between the same two points.

The Slope Formula

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

This formula essentially calculates the "rise over run" we talked about earlier. The numerator (y2 - y1) represents the change in the y-coordinates (the rise), and the denominator (x2 - x1) represents the change in the x-coordinates (the run).

Applying the Formula to Points A and B

Now that we have the formula, let's plug in the coordinates of our points A(4, 5) and B(9, 7). We'll label them as follows:

  • A(4, 5) => x1 = 4, y1 = 5
  • B(9, 7) => x2 = 9, y2 = 7

Substituting these values into the slope formula, we get:

m = (7 - 5) / (9 - 4)

Calculating the Slope

Let's simplify the expression:

m = 2 / 5

Therefore, the slope of line AB is 2/5 or 0.4. This means that for every 5 units you move to the right along the line, you move 2 units upwards. This positive slope confirms that the line is increasing as you move from left to right.

Visualizing the Line

Imagine plotting the points A(4, 5) and B(9, 7) on a coordinate plane. You can draw a line connecting these two points. The slope of 2/5 tells you how steep that line is. For every 5 units you move horizontally from point A, you must move 2 units vertically to reach another point on the line (in this case, point B).

Importance of Understanding Slope

Understanding the concept of slope is fundamental in various areas of mathematics and its applications. Here are a few reasons why:

  • Linear Equations: Slope is a key component of linear equations (y = mx + b), where 'm' represents the slope and 'b' represents the y-intercept. Knowing the slope allows you to define and analyze linear relationships.
  • Calculus: In calculus, the derivative of a function at a point represents the slope of the tangent line to the function's graph at that point. This is crucial for understanding rates of change and optimization problems.
  • Physics: Slope can represent physical quantities like velocity (the slope of a position-time graph) or acceleration (the slope of a velocity-time graph).
  • Real-World Applications: Slope is used in various real-world applications, such as determining the steepness of a road or ramp, calculating the pitch of a roof, or analyzing the rate of change in data.

Common Mistakes to Avoid

When calculating the slope, it's essential to avoid these common mistakes:

  • Incorrectly Subtracting Coordinates: Ensure you subtract the y-coordinates and x-coordinates in the same order. For example, always do (y2 - y1) / (x2 - x1) or (y1 - y2) / (x1 - x2), but don't mix the order.
  • Dividing by Zero: Remember that the denominator (x2 - x1) cannot be zero. If it is, the slope is undefined, and the line is vertical.
  • Mixing Up x and y: Always put the change in y-values (rise) over the change in x-values (run). Don't reverse them.

Conclusion

In summary, the slope of line AB, given points A(4, 5) and B(9, 7), is 2/5. We found this by using the slope formula m = (y2 - y1) / (x2 - x1) and substituting the coordinates of the points. Remember that understanding slope is crucial for working with linear equations, calculus, and various real-world applications. Now you have a solid understanding of how to calculate the slope of a line given two points! You can further explore linear equations and their properties by visiting Khan Academy's section on linear equations