Slope Problem: Finding 'r' In Points (r,-3) And (-3,-8)

Introduction to Slope and Coordinate Geometry

In the realm of coordinate geometry, understanding the concept of slope is fundamental. The slope of a line describes its steepness and direction, providing a crucial measure in various mathematical and real-world applications. When given two points on a line, we can determine the slope using a straightforward formula. However, what happens when one of the coordinates is unknown? This article delves into such a problem, guiding you through the process of finding an unknown coordinate using the slope formula. We'll explore a specific scenario where a line passes through the points (r, -3) and (-3, -8) and has a slope of -5. Our goal is to find the value of 'r'. This exercise not only reinforces the application of the slope formula but also highlights the importance of algebraic manipulation in solving geometric problems. This understanding is crucial for more advanced topics in mathematics and physics, where slopes represent rates of change and directional properties.

Understanding the Slope Formula

Before diving into the problem, it’s essential to have a solid grasp of the slope formula. The slope, often denoted as 'm', of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

This formula essentially calculates the “rise over run,” where the rise is the vertical change (difference in y-coordinates) and the run is the horizontal change (difference in x-coordinates). Understanding this formula is crucial because it provides a direct link between the coordinates of points on a line and the line's steepness. A positive slope indicates an upward inclination, a negative slope indicates a downward inclination, a slope of zero represents a horizontal line, and an undefined slope represents a vertical line. The slope formula is not just a mathematical tool; it's a fundamental concept that helps us visualize and analyze linear relationships, making it indispensable in fields like engineering, economics, and computer graphics.

Setting Up the Equation

In our problem, we are given two points (r, -3) and (-3, -8), and the slope of the line passing through these points is -5. To find the value of 'r', we will use the slope formula and substitute the given values. Let’s denote (r, -3) as (x₁, y₁) and (-3, -8) as (x₂, y₂). Then, we have:

x₁ = r, y₁ = -3 x₂ = -3, y₂ = -8 m = -5

Substituting these values into the slope formula, we get:

-5 = (-8 - (-3)) / (-3 - r)

This equation forms the basis for solving for 'r'. It's a crucial step because it translates the geometric problem into an algebraic one. The accuracy of this setup is paramount; a mistake here will propagate through the rest of the solution. By carefully substituting the given values, we create an equation that represents the relationship between the coordinates and the slope, allowing us to isolate and solve for the unknown variable 'r'.

Solving for 'r'

Now that we have the equation, the next step is to solve for 'r'. The equation we obtained from the previous step is:

-5 = (-8 - (-3)) / (-3 - r)

First, simplify the numerator:

-5 = (-8 + 3) / (-3 - r) -5 = -5 / (-3 - r)

Next, to isolate the term with 'r', multiply both sides of the equation by (-3 - r):

-5 * (-3 - r) = -5

Distribute the -5 on the left side:

15 + 5r = -5

Now, subtract 15 from both sides:

5r = -5 - 15 5r = -20

Finally, divide both sides by 5 to solve for 'r':

r = -20 / 5 r = -4

Therefore, the value of 'r' is -4. This step-by-step algebraic manipulation is crucial for arriving at the correct solution. Each step, from simplifying the numerator to isolating 'r', must be performed with precision. This process not only provides the answer but also reinforces the importance of algebraic skills in solving geometric problems. The ability to manipulate equations and isolate variables is a fundamental skill in mathematics and its applications.

Verification and Conclusion

To ensure our solution is correct, it's always a good practice to verify the answer. We can do this by substituting the value of 'r' we found back into the original slope formula. If r = -4, our points are (-4, -3) and (-3, -8). The slope 'm' would then be:

m = (-8 - (-3)) / (-3 - (-4)) m = (-8 + 3) / (-3 + 4) m = -5 / 1 m = -5

Since the calculated slope matches the given slope of -5, our solution is verified. This verification step is a crucial part of the problem-solving process. It confirms that our algebraic manipulations were correct and that the value of 'r' we found satisfies the given conditions. In conclusion, the value of 'r' for which the line passing through the points (r, -3) and (-3, -8) has a slope of -5 is -4. This problem illustrates the application of the slope formula and the importance of algebraic skills in solving geometric problems. The ability to find unknown coordinates using slopes is a valuable tool in mathematics and related fields, providing a foundation for understanding linear relationships and their properties.

For further exploration of slope and linear equations, you might find valuable resources on websites like Khan Academy's Algebra I section.