Problem 31 - College Algebra with Corequisite Support

Question

For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line). $$ (-1,2)(0,4)(2,1) $$

Step-by-Step Solution

Answer

The points are not collinear.

Step 1: Understand the Problem

We need to plot three points on a coordinate plane: Point A: \((-1,2)\), Point B: \((0,4)\), Point C: \((2,1)\). After plotting, we will determine if these points are collinear; that is, if they lie on a single straight line.

Step 2: Plot Point A

Point A is \((-1,2)\). To plot this, start at the origin (0,0) on the coordinate plane. Move 1 unit to the left along the x-axis and 2 units up along the y-axis. Mark this point.

Step 3: Plot Point B

Point B is \((0,4)\). This point lies on the y-axis because the x-coordinate is 0. Start at the origin (0,0) and move 4 units up. Mark this point.

Step 4: Plot Point C

Point C is \((2,1)\). Start again at the origin (0,0). Move 2 units to the right along the x-axis and 1 unit up along the y-axis. Mark this point.

Step 5: Check for Collinearity

To determine if the points are collinear, calculate the slope between pairs of points and see if they are equal. - Slope between Point A (-1,2) and Point B (0,4): \[\frac{4-2}{0 - (-1)} = \frac{2}{1} = 2\]- Slope between Point B (0,4) and Point C (2,1): \[\frac{1-4}{2-0} = \frac{-3}{2}\]- Slope between Point A (-1,2) and Point C (2,1): \[\frac{1-2}{2-(-1)} = \frac{-1}{3}\]Since the slopes are not equal, the points are not collinear.

Key Concepts

Points Plotting

In the world of coordinate geometry, plotting points on a plane is essential. Each point is defined by a pair of coordinates \((x, y)\). Here's a simple process to plot points on the coordinate plane:

• Identify the Coordinates: Each point has an \(x\)-coordinate and a \(y\)-coordinate. The \(x\)-coordinate tells you how far to move horizontally. The \(y\)-coordinate tells you how far to move vertically.
• Start at the Origin: Begin at the origin, which is (0,0), where the x-axis and y-axis intersect.
• Move According to Coordinates: For the point \((-1,2)\), move 1 unit left and 2 units up. For \((0,4)\), move 4 units straight up from the origin. For \((2,1)\), move 2 units right and 1 unit up.
• Mark the Point: Once you've moved the correct distances along the x and y axes, mark the point.

Plotting points correctly ensures the accuracy of any subsequent analysis. It’s a foundational skill that makes understanding geometry much simpler.

Collinearity

Collinearity is a concept that indicates whether points lie on a single straight line.

• Visual Inspection: Once you have plotted the points, a quick visual inspection can give you a rough idea if they look aligned.
• Mathematical Approach: Calculate the slopes between each pair of points to verify if they are equal. If all pairs have the same slope, the points are collinear.
• Practical Example: For our given points, the slopes were calculated as follows: between the first and second points as 2, the second and third points as \(-1.5\), and the first and third points as \(-0.33\).

Since these slopes are not identical, visually or mathematically, they do not indicate collinearity. Knowing if points are collinear helps in understanding underlying patterns and relationships in data.

Slope Calculation

The slope of a line is a measure of its steepness, often important in geometry and algebra. Here's how you calculate it:

• Formula: The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• Slope Interpretation: A positive slope indicates an upward direction from left to right, while a negative slope indicates a downward direction.
• Finding Slopes: In our scenario, the slope between \((-1,2)\) and \((0,4)\) is 2. The slope between \((0,4)\) and \((2,1)\) is \(-1.5\), and between \((-1,2)\) and \((2,1)\) is \(-0.33\).

Understanding slope is critical, especially when determining if points are collinear. Different slopes clearly show varying angles and therefore no linear alignment across our points.