Problem 30 - College Algebra

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). $$(4,1)(-2,-3)(5,0)$$

Step-by-Step Solution

Answer

The points are not collinear.

Step 1: Plot the Points

Start by plotting the points \((4,1)\), \((-2,-3)\), and \((5,0)\) on a coordinate plane. Place point \((4,1)\) on the plane by moving 4 units to the right on the x-axis, and 1 unit up on the y-axis. Next, plot point \((-2,-3)\) by moving 2 units left and 3 units down. Finally, place point \((5,0)\) by moving 5 units to the right and staying on the x-axis (since y is 0).

Step 2: Inspect Collinearity Visually

Check if all three points lie on a straight line by visually inspecting the plotted points. Draw a straight line connecting two of the points and see if the third point lies exactly on this drawn line.

Step 3: Check Collinearity Algebraically

To confirm if the points are collinear, calculate the slopes between pairs of points and see if they are equal. Calculate the slope between \((4,1)\) and \((-2,-3)\):\[ m_{1} = \frac{1 - (-3)}{4 - (-2)} = \frac{4}{6} = \frac{2}{3} \]Calculate the slope between \((4,1)\) and \((5,0)\):\[ m_{2} = \frac{1 - 0}{4 - 5} = \frac{1}{-1} = -1 \]Calculate the slope between \((-2,-3)\) and \((5,0)\):\[ m_{3} = \frac{-3 - 0}{-2 - 5} = \frac{-3}{-7} = \frac{3}{7} \]Since none of the slopes are equal, the points are not collinear.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and shapes. It is defined by two perpendicular lines known as the axes. The horizontal axis is called the x-axis, and the vertical axis is the y-axis. The point where these axes intersect is the origin, denoted as \(0,0\).
The coordinate plane is divided into four quadrants. Each quadrant has different signs for x and y coordinates:

• Quadrant I: (+x, +y)
• Quadrant II: (-x, +y)
• Quadrant III: (-x, -y)
• Quadrant IV: (+x, -y)

You can picture the coordinate plane like a map. By using this map, you can find where any specific point is located based on its coordinates. For instance, the point \(4,1\) is located by moving 4 units to the right on the x-axis and 1 unit up on the y-axis.

Plotting Points

Plotting points on the coordinate plane is an essential skill in math, helping you to visualize coordinates and their relationships. To plot a point, you begin at the origin \(0,0\). For a coordinate, such as \(4,1\), move right on the x-axis by 4 units, and up 1 unit on the y-axis. This will place you directly at the point.
Here’s how you can plot the given points:

• For \(4, 1\): Move 4 units to the right and 1 unit up.
• For \(-2, -3\): Move 2 units to the left and 3 units down.
• For \(5, 0\): Move 5 units to the right and remain on the x-axis as there’s no movement on the y-axis.

Each coordinate forms a point on the plane. Connecting these plotted points visually helps to determine if they align in a straight line, which leads to examining their collinearity.

Slope Calculation

The concept of a slope is crucial in determining the relationship between points. The slope represents the rate of change of the y-coordinate with respect to the x-coordinate. For a straight line, slope is calculated by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Slope helps in assessing the steepness of a line. When plotting multiple points, you can calculate the slope between pairs of points to determine if they are collinear. Points are collinear if they lie on the same straight line, which means all calculated slopes between points must be equal.
As an example, consider three points; calculate slopes between each pair. If any pair does not share the same slope, then the points are not collinear. In our example, slopes: \(\frac{2}{3}\), -1, and \(\frac{3}{7}\) are found. Since these slopes are different, the points \(4,1\), \(-2,-3\), and \(5,0\) are not collinear.