Problem 66 - Algebra 1

Question

When you use the formula \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) to find the slope of a line passing through two points, does it matter which point you choose to use as \(\left(x_{1}, y_{1}\right)\) ? Give three different examples to support your answer.

Step-by-Step Solution

Answer

No, it doesn't matter which point is chosen as \((x_{1}, y_{1})\) or \((x_{2}, y_{2})\) while calculating the slope of the line passing through given points. This is supported by the three examples where swapping points didn't change the calculated slope.

Step 1: Example 1

Consider two points A(2,3) and B(5,7). First, use A as (x1,y1) and B as (x2,y2), the slope m is \(\frac{7-3}{5-2} = \frac{4}{3}\). Then switch the points such that B is (x1,y1) and A is (x2,y2), the slope is \(\frac{3-7}{2-5} = \frac{4}{3}\). The two slopes are equal, suggesting that the choice of points does not affect the slope.

Step 2: Example 2

Consider two points C(1,2) and D(6,8). First, use C as (x1,y1) and D as (x2,y2), the slope m is \(\frac{8-2}{6-1} = \frac{6}{5}\). Then switch the points such that D is (x1,y1) and C is (x2,y2), the slope is \(\frac{2-8}{1-6} = \frac{6}{5}\). The two slopes are equal, suggesting that the choice of points does not affect the slope.

Step 3: Example 3

Consider two points E(4,-1) and F(7,0). First, use E as (x1,y1) and F as (x2,y2), the slope m is \(\frac{0-(-1)}{7-4} = \frac{1}{3}\). Then switch the points such that F is (x1,y1) and E is (x2,y2), the slope is \(\frac{-1-0}{4-7} = \frac{1}{3}\). The two slopes are equal, suggesting that the choice of points does not affect the slope.

Key Concepts

points

In coordinate geometry, points are fundamental units used to define location on a two-dimensional plane. A point consists of two numbers, which are usually referred to as coordinates. These coordinates are written as \(x, y\), where \(x\) is the horizontal position, and \(y\) is the vertical position of the point. Understanding how to work with points is essential for analyzing geometric shapes and calculating various properties like slopes or distances.

When working with points to find the slope between them, it’s important to assign these coordinates consistently based on the given context. However, as demonstrated in the examples from the original exercise, the choice of which point is \( (x_1, y_1) \) and which is \( (x_2, y_2) \) does not affect the outcome for calculating a slope. By switching the roles of the points in the examples, we observed that the slope value remained unchanged, illustrating the concept's flexibility.

coordinate geometry

The field of coordinate geometry, also known as analytic geometry, involves using an algebraic approach to solve geometric problems. It combines algebra and geometry by plotting points on the Cartesian plane, which is a scaled grid defined by horizontal (x-axis) and vertical (y-axis) lines.

In the context of the slope calculation, coordinate geometry helps us to represent geometric figures algebraically and perform calculations accurately. The formula \( \frac{y_2 - y_1}{x_2 - x_1} \) allows us to find the slope of a line passing through any two given points on this Cartesian plane.

This mathematical representation makes it easier to analyze and visualize relationships between different geometrical entities. By transitioning between graphical and algebraic expressions quickly, coordinate geometry forms a bridge connecting these concepts comprehensively.

mathematical proof

Mathematical proofs are logical arguments that verify the truth of a statement or proposition using deductive reasoning. They are crucial for confirming that mathematical concepts and equations hold universally, for all possible scenarios.

In the given exercise, a form of mathematical proof is demonstrated through repetition and consistent results. By taking multiple pairs of points and calculating the slopes, we established that swapping the points does not change the slope. This empirical evidence acts as an informal proof to assert that the order of choosing which point is \( (x_1, y_1) \) or \( (x_2, y_2) \) in the slope formula is irrelevant.

While this type of proof is more observational and relies on patterns seen through specific examples, formal proofs would involve abstract verification. Nevertheless, both informal and formal proofs serve as robust tools in mathematics to establish the correctness and validity of concepts across various applications.