Problem 73 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \frac{7+3 i}{5+3 i}=\frac{7}{5} $$

Step-by-Step Solution

Answer

The statement is True.

Step 1: Analyze the Statement

In an x-y plane, a point on the y-axis has its x-coordinate as 0. That's because any point directly on the y-axis is not moved left or right from the origin, therefore its x-coordinate (representing horizontal movement) is 0.

Step 2: Comparing the Statement with the Fact

The statement given in the exercise is 'If a point is on the y-axis, its x-coordinate must be 0.' This statement is indeed a True statement. No changes are required to make the statement True.

Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system is a foundational element in the study of geometry and algebra, constituting a plane composed of two perpendicular axes: the horizontal axis, often referred to as the x-axis, and the vertical axis, called the y-axis. These axes intersect at a point known as the origin, labeled as (0,0).

Every point on this plane can be represented by a pair of numerical coordinates, which are essentially distances from the origin along the x-axis and y-axis. The coordinates are written in the form (x, y), wherein the first number represents the x-coordinate, or the horizontal distance from the origin, and the second, the y-coordinate, or the vertical distance.

Understanding this system is crucial as it allows us to describe any location within a two-dimensional space accurately. For example, the coordinate (3, -2) signifies a point located 3 units to the right and 2 units down from the origin.

X-coordinate

In the realm of coordinate geometry, the x-coordinate plays a significant role, denoting the position of a point along the horizontal axis in the Cartesian coordinate system. It reflects how far a point is from the y-axis, which vertically slices the plane into two symmetrical halves.

To visualize this, imagine walking straight ahead or backward from the center of a crosswalk—the x-coordinate embodies those steps you take. If you move right from the starting point, the origin, the x-coordinate is positive, but moving left gives the x-coordinate negative values. It is the first number in the ordered pair that constitutes the coordinates of a point.

For instance, in the coordinate (5,7), the x-coordinate is 5, suggesting the point lies 5 units to the right of the y-axis. It's vital to discern the role of the x-coordinate, as it lays the groundwork for understanding the positioning and movement of points on a horizontal scale.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry, using the principles of both to solve geometrical problems with algebraic equations. This field is particularly adept at addressing problems around the shapes, sizes, and positions of figures on the Cartesian plane.

By assigning numerical coordinates to points and employing algebraic formulas, one can calculate distances, angles, and other properties of geometric shapes like lines, circles, and polygons. For example, we can use the distance formula to calculate the distance between two points or the slope formula to find the incline of a line. It's an essential part of mathematics because it provides a bridge between abstract algebraic concepts and real-world spatial understanding.

Moreover, coordinate geometry plays a crucial role in fields beyond pure mathematics, such as physics, engineering, computer graphics, and navigation, proving its significance in both theoretical and practical domains.