Problem 15 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \(\left(-2,-3 \frac{1}{5}\right)\)

Step-by-Step Solution

Answer

The short answer consists of the position of the point. In a rectangle coordinate system or Cartesian coordinate system, the point (-2, -3 1/5) is located 2 units to the left and 3.2 units down from the origin, placing it in the 3rd Quadrant.

Step 1: Identify the x coordinate

The x-coordinate of the given point (-2, -3 1/5) is -2. This suggests that the point is located 2 units to the left of the origin.

Step 2: Identify the y coordinate

The y-coordinate of the given point (-2, -3 1/5) is -3 1/5. To accurately plot this point, convert this mixed number to an improper fraction (-16/5). In a normal circumstance where 1 unit on y-axis represent 1, this suggests that the point is located 3.2 units below the origin. This is because 16 divided by 5 is 3.2.

Step 3: Plotting the Point

Starting from the origin, move 2 units to the left (as x = -2) along the x-axis and then 3.2 units down along the y-axis (as y = -3.2) to reach the correct location of the point. Place a dot on this location and label it as (-2, -3 1/5), this is your required point.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system is a method used to pinpoint the location of a point in a plane using two numbers, known as coordinates. This system is often visualized as a grid.

• The horizontal line is called the x-axis.
• The vertical line is called the y-axis.
• The point where these axes intersect is known as the origin, represented by (0,0).

Each point is identified by an x and y coordinate, usually written in the form (x, y).

In this context, the coordinates (-2, -3 1/5) indicate the specific position on the plane. The x-coordinate, -2, determines the horizontal movement, while the y-coordinate, -3 1/5, determines the vertical movement. Understanding this system is essential for plotting points accurately.

Plotting Points

Plotting points in the rectangular coordinate system is like finding a treasure on a map. You use the coordinates to determine exactly where to place the point.

How to Plot

• Start at the origin (0,0).
• Move horizontally according to the x-coordinate. If it's positive, go right. If negative, go left.
• Move vertically according to the y-coordinate. If it's positive, go up. If negative, go down.
• Place a dot where these two movements intersect.

For the point (-2, -3 1/5):

• Begin at the origin.
• Move 2 units left (because of the x-coordinate -2).
• Next, move 3.2 units down (because of the y-coordinate -3 1/5 converted to -3.2).

Mark this spot and label it. This process allows you to locate any point in the coordinate system.

Mixed Numbers

A mixed number consists of a whole number and a fraction, like -3 1/5 in the exercise. Understanding how to work with them is crucial for accuracy, especially in plotting.

Converting Mixed Numbers

To plot a point like (-2, -3 1/5) on a coordinate grid, it's often useful to convert the mixed number into an improper fraction or a decimal.

• The whole part (-3) and the fractional part (1/5) combine to form the mixed number.
• In mathematical terms, convert it:
  Multiply the whole number by the denominator: 3 x 5 = 15.
  Add the numerator: 15 + 1 = 16.
• Thus, -3 1/5 becomes -16/5 as an improper fraction.
• As a decimal, this becomes -3.2 (since 16 divided by 5 equals 3.2).

Understanding these conversions helps in the precise placement of points on the grid.