Question

The ordinate of any point on a certain straight line is \(-5 .\) Give the coordinates of the point of intersection of that line and the \(y\) axis.

Short answer

The coordinates of the point of intersection of the line and the y-axis are (0, -5).

Step-by-step solution

Understanding the ordinate of a point

The ordinate of a point refers to the second element in its ordered pair, or its coordinate on the y-axis. In this case, any point on the line has an ordinate of -5, meaning its y-coordinate is -5 regardless of the x-coordinate.

Finding the intersection with the y-axis

Since the line intersects the y-axis, at this point the x-coordinate of any point on the line must be 0. Hence, the coordinates of the intersection point will be (0, y).

Determine the y-coordinate of the intersection point

Given that the ordinate is always -5, and the x-coordinate at the y-axis must be 0, the coordinate of the point of intersection is (0, -5).

Key concepts

Cartesian Coordinate System

The Cartesian coordinate system is a foundational concept in mathematics and by extension is critical in various fields such as physics, engineering, and computer science. The system is composed of two number lines perpendicular to each other, labeled as the x-axis (horizontal line) and y-axis (vertical line). The point where they intersect is known as the origin, denoted as (0, 0).

This system allows us to specify the location of points in a two-dimensional space using a pair of numerical coordinates. The first number is called the abscissa and refers to the position along the x-axis, while the second number is the ordinate, indicating the position along the y-axis. These coordinates are often written in the form \( (x, y) \), known as an ordered pair. The entire plane is divided into four sections called quadrants, where the sign of the coordinates changes depending on the quadrant the point lies in. This structure is essential for graphing equations and interpreting geometric shapes and their properties within two dimensions.

Y-axis Intersection

The concept of a y-axis intersection concerns the point where a line or curve crosses the y-axis. This point is of particular interest because it reveals a consistent property of any graph in Cartesian coordinates: at the y-axis, the x-coordinate of any point is always 0. This fact is key to identifying the point where a line intersects the y-axis, as only the y-coordinate will differ.

To find the y-axis intersection of a straight line, we can look at the line equation or plot it graphically. But even without the full line equation, knowing that the ordinate is consistent allows us to determine the point of intersection with the y-axis, a concept that can be applied to solve problems in algebra and geometry with ease. For instance, in the provided exercise, it was deduced that the y-axis intersection of the line with an ordinate of -5 is at the point (0, -5), because at the y-axis, the x-coordinate must be 0, and we already know the line's ordinate.

Ordered Pairs

An ordered pair is a set of two elements where the order in which the elements appear is significant. In the Cartesian coordinate system, each point is represented by an ordered pair \( (x, y) \), where \( x \) signifies the point's horizontal position, and \( y \) its vertical position. The name 'ordered pair' comes from the importance of the sequence; switching \( x \) and \( y \) would refer to a completely different point in the plane.

In the context of our exercise, \( (-5) \) is the ordinate of a point on a straight line, suggesting that for every point on this line, the second element in the ordered pair is \( -5 \). Despite the x-coordinate (the abscissa) varying, the y-coordinate remains consistent. When we talk about the y-axis intersection, we are referring to the special case where the ordered pair takes the form of \( (0, y) \), indicating the line's crossing point with the y-axis. Ordered pairs serve not just in positioning but also in the defining functions, relationships between sets, and in the broader realm of discrete mathematics.