Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

Plot the given point in a rectangular coordinate system. $$\left(-\frac{5}{2}, 0\right)$$

Step-by-Step Solution

Answer

The point \(-\frac{5}{2}, 0\) lies 2.5 units to the left of the origin on the x-axis.

Step 1: Understand the coordinate system

Many algebraic problems use a rectangular (or Cartesian) coordinate system to understand the relationship between two variables. This system comprises two perpendicular lines (or axes): the x-axis (horizontal line) and the y-axis (vertical line). The point where these two lines intersect is the origin, or the point (0,0).

Step 2: Identify coordinates

Coordinates are given in ordered pairs (x, y). The initial number refers to the x-coordinate (or how far to move left or right) and the second number represents the y-coordinate (how far to move up or down). In this case, the given point has coordinates \(-\frac{5}{2}\) (x-coordinate), 0 (y-coordinate). We start at the origin (0,0).

Step 3: Plotting x-coordinate

Now, move along the x-axis. A negative x-coordinate means we move to the left. Since our x-coordinate is \(-\frac{5}{2}\), we move 2.5 units to the left of the origin.

Step 4: Plotting y-coordinate

Next is the y-coordinate. It tells us how far to move up (+) or down (-) the y-axis. The given y-coordinate is 0, which means we stay on the x-axis and we don't move up or down.

Step 5: Mark the Point

At the required position on the x-axis, mark the point and it will be the given point. In this case, the point \(-\frac{5}{2}, 0\) will be located 2.5 units to the left of the origin on the x-axis.

Key Concepts

Ordered Pairs

Ordered pairs are the coordinates of points in the coordinate system, and they always come in a specific sequence: (x, y). Like a treasure map, they give you the X marks the spot, followed by a step north or south.

Here's a simple guide:

• Think of the first number as how many paces you take left or right from the center—the origin.
• The second number tells you how many steps to walk up or down from that new spot.
• Together, they lead you to the exact point where X (the x-coordinate) meets Y (the y-coordinate)—the treasure!
• Remember, the order matters; swapping the numbers lands you at a completely different spot!

That's why they're called 'ordered' pairs – keep them in order and you'll find your mathematical treasure.

X-axis and Y-axis

The x-axis and y-axis are the foundational lines in the coordinate system, akin to the main roads in a city grid. They intersect at a central point, the origin, which is the starting point for all journeys within the coordinate plane.

An Easy Walkabout on the Axes

Strolling along the x-axis takes you left or right, while a jaunt up or down happens on the y-axis. Here are the basic directions:

• The x-axis is horizontal – picture it as the ground beneath your feet, with left-hand steps (negative) and right-hand steps (positive).
• The y-axis stands tall and vertical – like a ladder reaching to the stars. Step up for positive values, or down for negative ones.
• Every point dances to the tune of these axes; their numbers lead you to the perfect spot.
• Together, they create a grid where each square can be a unique address on the map of mathematics.

Logically navigating the x-axis and y-axis can turn the maze of numbers into a well-charted cityscape!