Problem 32 - 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). $$ (-3,0)(-3,4)(-3,-3) $$

Step-by-Step Solution

Answer

The points are collinear as they lie on the vertical line x = -3.

Step 1: Understand the Points

The points given are 1. (-3, 0) 2. (-3, 4) 3. (-3, -3) These points are given in a two-dimensional Cartesian coordinate system and each point has coordinates in the form (x, y).

Step 2: Plot Each Point

To plot the points, consider each point's coordinates: 1. (-3, 0): Move 3 units left from the origin (0,0) to the x-coordinate of -3, and since the y-coordinate is 0, the point is on the x-axis. 2. (-3, 4): Move 3 units left to the x-coordinate -3, then move 4 units up for the y-coordinate of 4. 3. (-3, -3): Move 3 units left to the x-coordinate -3, then move 3 units down for the y-coordinate of -3.

Step 3: Analyze the Positions of the Points

Now that you've plotted the points, observe their alignment. All points have the same x-coordinate, -3, meaning they are vertically aligned.

Step 4: Verify Collinearity

Since all three points have the same x-coordinate, they lie on the same vertical line, x = -3. This confirms that the points are collinear, as they lie on the same straight line.

Key Concepts

Cartesian Coordinate System

The Cartesian Coordinate System is a two-dimensional grid that allows us to visualize and locate points using two numbers - usually called coordinates. This system was named after the French philosopher and mathematician René Descartes.

The coordinate plane is made up of two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis).

• The point where these two axes intersect is called the origin, denoted by \( (0,0) \).
• Coordinates are written as an ordered pair \( (x, y) \), indicating the horizontal and vertical distances from the origin.

This system is instrumental in representing and solving geometric problems. Knowing the position of points in the plane through their coordinates can help us determine their relationship with each other, such as identifying if they are collinear.

Plotting Points

Plotting points in the Cartesian coordinate system involves determining the position of each point using its coordinates. Begin by considering each ordered pair separately. Each point gives specific instructions on how to navigate the grid.

• For a point like \( (-3, 0) \), the x-value of -3 tells us to move 3 units to the left from the origin, while the y-value 0 signifies no vertical movement.
• With \( (-3, 4) \), after moving 3 units left, move 4 units up since the y-coordinate is 4.
• Lastly, for \( (-3, -3) \), move 3 units left and 3 units down, as negative values direct us downwards.

By following these movements, you will accurately place each point on the graph. This is a crucial skill in geometry and algebra, enabling you to visualize the spatial relationship between points.

Vertical Line

A vertical line is a straight line that goes up and down in a Cartesian coordinate system. Unlike horizontal lines, which respond to changes in the y-value, vertical lines maintain the x-coordinate. Such lines are represented by equations where x is a constant, like \( x = -3 \).

• If all plotted points have the same x-coordinate, they are aligned vertically.
• In our example, all points (\(-3, 0\), \(-3, 4\), \(-3, -3\)) have an x-coordinate of -3, confirming they lie on the vertical line \( x = -3 \).
• This condition of equal x-coordinates across points establishes "collinearity," meaning the points are on the same line.

Understanding vertical lines is crucial when determining the nature of point alignments and predicting intersections with other lines, but also recognizing foundational directional behaviors in geometry.